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ARITHMETIC 


BY 

EUGENE  HERZ 

CERTIFIED    PUBLIC   ACCOUNTANT 
AND 

MARY  G.  BRANTS 

CRITIC   TEACHER,    PARKER  PRACTICE  SCHOOL, 
CHICAGO  NORMAL  COLLEGE 

WITH   THE   EDITORIAL  ASSISTANCE   OF 

GEORGE  GAILEY  CHAMBERS,  Ph.D. 

ASSISTANT  PROFESSOR   OF  MATHEMATICS, 
UNIVERSITY   OF   PENNSYLVANIA 


PARTS   VII  AND   VIII 

ADVANCED  LESSONS 


THE  JOHN  C.  WINSTON  COMPANY 
CHICAGO  PHILADELPHIA  Toronto 


irtT' 


Copyright,  1920,  by 
1'he  Joh^  C.  Winston  Company 


Filtered  at  Stationers'  Hall,  London 
All  rights  reserved 


FOREWORD 

Bearing  in  mind  that  a  thorough  knowledge  of  arithmetic 
is  perhaps  more  frequently  the  cause  of  success  in  hfe  than  is 
any  other  single  factor,  one  can  hardly  overestimate  the 
importance  of  this  subject  to  the  future  welfare  of  the  child, 
nor  can  one  fail  to  reahze  how  great  is  the  responsibility 
which  rests  on  those  whose  duty  it  is  to  provide  for  his  edu- 
cation in  this  branch. 

No  book  or  series  of  books  can  possibly  illustrate  every 
use  to  which  numbers  can  be  put,  but  if  the  principles  under- 
lying their  use  are  properly  taught,  the  child  can  reason  for 
himself  the  proper  application  of  his  knowledge  to  any  given 
problem.  Furthermore,  as  he  must  know  not  merely  how 
to  solve  a  problem,  but  how  to  solve  it  in  the  quickest  and 
simplest  manner,  he  must  know  not  merely  the  various  proc- 
esses, but  their  construction  as  well;  he  must  be  able  to 
analyze  to  such  an  extent  that  when  a  problem  is  presented 
to  him,  he  can  distinguish  the  facts  which  are  relevant 
from  those  which  are  irrelevant,  he  can  separate  the  known 
from  the  unknown,  he  can  arrange  the  known  in  logical  order 
for  his  processes,  and  he  can  use  the  shortest  processes  pos- 
sible. An  attempt  to  give  the  pupil  this  ability  is  the  motive 
for  this  work. 

The  vehicle  used  to  obtain  the  result  is  a  series  of  pro- 
gressive lessons,  which,  with  ample  practice,  take  the  pupil 
step  by  step  through  the  construction  of  each  process  to  be 
learned,  thus  giving  him  the  opportunity  of  following  the 
teacher's  explanation,  and  of  referring  to  past  lessons  at  any 
time.  In  this  way  the  pupil  who  is  slower  to  grasp  new  ideas 
than  the  average  can  keep  up  with  his  class,  and  every  pupil 
can  at  all  times  refresh  his  memory  on  any  points  which  he 
may  have  forgotten  or  which  may  have  escaped  him  in  the 
classroom,  and  which  have  so  often  been  lost  to  him  forever. 

464448 


The  time-saving  methods  used  by  the  most  expert  arithme- 
ticians are  introduced  as  part  of  the  routine  work;  thus, 
these  become  a  part  of  the  child's  general  education  without 
any  special  effort  on  his  part. 

It  is  not  intended  that  the  lessons  or  definitions  are  to  be 
learned  verbatim,  any  more  than  it  is  intended  that  the 
examples  given  are  to  be  memorized;  both  are  there  for  the 
purpose  of  showing  the  pupil  the  reason  for,  and  the  applica- 
tion of,  the  processes,  and  the  exercises  are  there  to  give 
him  practice  and  to  test  his  knowledge  of  what  he  has  learned. 

The  exercises  are  prepared  in  such  manner  that  they  form 
an  automatic  and  continuous  review  of  what  has  been  learned, 
but  further  review  work  is  given  at  regular  intervals. 

The  series  consists  of  Three  Books  and  Teacher's  Manuals, 
as  follows: 

Primary  Lessons Parts    I    and    II.     (Teacher's 

Manual  only.) 

Elementary  Lessons. .  .  .  Parts  III  and  IV.  (With  Manu- 
al for  the  Teacher.) 

Intermediate  Lessons.. .  Parts  V  and  VI.    (With  Manual 

for  the  Teacher.) 

Advanced  Lessons Parts    VII    and    VIII.      (With 

Manual  for  the  Teacher.) 

The  first  two  parts  are  so  arranged  in  the  Teacher's  Manual 
that  the  lessons  and  exercises  can  be  given  largely  as  games, 
play  work,  number  stories,  in  language  work,  etc.,  all  used 
more  or  less  incidentally,  till  the  child  is  gradually  prepared 
for  work  requiring  an  increasing  degree  of  conscious  effort. 

The  work  contained  in  each  of  the  eight  parts  is  that 
which  is  usually  taught  in  the  corresponding  grade,  and  it  is 
recommended  that  this  routine  be  followed.  However,  spe- 
cial provision  has  been  made  for  such  variations  in  the 
grading  as  are  required  in  some  localities,  by  means  of  a 
series  of  notes  in  the  Teacher's  Manuals  which  enable  the 
teacher  to  follow  either  method  with  equal  facility. 


CONTENTS 

PART  vn 

LESSON 

NUMBER  PAGE 

Denominate  Numbers 

1.  Reduction 1 

2.  Reduction  of  Fractional  Denominate  Numbers 4 

3.  Addition  of  Denominate  Numbers 6 

4.  Subtraction  of  Denominate  Numbers 11 

5.  Multiplication  of  Denominate  Numbers 12 

6.  Division  of  Denominate  Numbers 14 

7.  Measuring  Land 20 

8.  Paper  Measure 24 

9.  Printers'  Type  Measure 27 

10.  Legal  Weights  of  a  Bushel  (In  Pounds) 33 

11.  Special  Working  Units 41 

Fractions 

12.  Compound  and  Complex  Fractions 46 

Multiplication 

13.  Cross  Multiplication 50 

Time  and  Wages 

14.  How  Wages  Are  Figured 54 

15.  Transposition  in  Figuring  Wages 59 

Mensuration 

16.  The  Circle 66 

17.  The  Ratio  of  the  Circumference  to  the  Diameter.  ...  67 

18.  Finding  the  Area  of  a  Circle 70 

19.  Finding  the   Area  of  the   Surface   of   a  Right   (Rect- 

angular) Prism 75 

20.  Finding  the  Area  of  the  Surface  of  a  Cylinder 77 

21.  Cutting  Material  to  Avoid  Waste 81 

22.  Finding  the  Volume  of  a  Cylinder 84 


CONTENTS 

LESSON 

NUMBER  PAGE 

Percentage 

23.  Successive  Trade  Discounts 91 

24.  Finding  the  Gross  Amount  When  the  Rates  of  Discount 

AND  the  Net  Amount  Are  Given 95 

25.  Insurance 97 

26.  Commission  and  Brokerage 100 

27.  Taxes 105 

28.  Computing  Interest  When  There  Are  Partial  Payments  .  109 

29.  Finding  the  Principal  When  the  Time,  Rate,  and  Interest 

Are  Given 113 

.30.  Finding  the  Time  When  the  Principal,  Rate,  and  Interest 

Are  Given 115 

31.  Finding  the  Rate  When  the  Principal,  Time,  and  Interest 

Are  Given 117 

32.  Transposition  in  Figuring  Interest 120 

33.  Compound  Interest 122 

Accounts 

34.  Savings  Bank  Accounts 125 

35.  Bank  Accounts  which  Are  Subject  to  Check 129 


CONTENTS 
PART  vm 

LESSON 

NUMBER  PAGE 

Notation  and  Numeration* 

1.  The  Higher  Periods 1 

Denominate  Numbers 

2.  Table  of  Circular  Measure 5 

3.  Longitude  and  Time 8 

4.  Standard  Time  in  the  United  States 13 

5.  The  International  Date  Line 16 

6.  The  Metric  System 18 

7.  Foreign  Money 23 

Powers  and  Roots 

8.  What  Powers  Are — Squaring  and  Cubing 31 

9.  What  Roots  Are 33 

10.  How  TO  Extract  the  Square  Root.     (Integers) 34 

11.  How    to    Extract    the    Square    Root.       (Decimals    and 

Fractions) 38 

12.  How  to  Extract  the  Square  Root  by  Factoring 41 

Equations 

13.  Numbers  and  Quantities  Represented  by  Letters 45 

14.  Solving  Equations.     (Adding  and  Subtracting) 47 

15.  Solving  Equations.     (Multiplying  and  Dividing) 49 

Mensuration 

16.  Right  Triangles 53 

17.  Isosceles  and  Equilateral  Triangles 57 

18.  Similar  Triangles 60 

19.  Table  of  Angular  Measure 68 

20.  Measuring  the  Length  of  Arcs  and  the  Area  of  Sectors 

OF  Circles 70 

21.  Pyramids 74 


1 


CONTENTS 

LESSON 

number  page 

22.  Cones 80 

23.  Frustums  (For  Surface  Work  Only) 84 

24.  Spheres 88 

Graphic  Charts  and  Meters 

25.  Graphic  Charts 06 

26.  Meters 102 

Percentage 

27.  Interest  on  Installment  Accounts 107 

28.  Bank  Discount Ill 

29.  Mortgages  and  Bonds 115 

30.  Corporations  and  Their  Capital  Stock 120 

31.  Rate  of  Income  (Yield)  on  Stocks  and  Bonds  Bought  at 

A  Premium  or  Discount 127 

32.  Insurance 134 

Partnership 

33.  Division  of  Profits  and  Losses 14^ 

Definitions  of  the  Terms  Used  in  Parts  I  to  VIII,  Inclusive.  153 
Abbreviations  and  Signs  Used  in  Parts  I  to  VIII,  Inclusive.  . .  169 


Tables  of  Weights  and  Measures 

(For  Ready  Reference) 
Dry  Measure 


I 

I  PINT       I  QUART      I  BUSHEL  I    PECK 

2  pints  (pt.) =1  quart  (qt.) 

8  quarts =  1  peck  (pk.) 

4  pecks =  1  bushel  (bu.) 


Avoirdupois  Weight 


16  ounces  (oz.) =1  pound  (lb.) 

100  pounds =1  hundredweight  (cwt.) 

20  hundredweight =1  ton  (T.) 

2,000  pounds =1  short  ton 

2,240  pounds =1  long  ton  (used  at  mines 

and  U.  S.  Custom  House) 


TABLES  OF  T^HEIGHTS  AND  MEASURES 


Linear  Measure 


12    inches  (in.) ......  =  1  foot  (ft.) 

3    feet p  1  yard  (yd.) 

5i  yards =  1  rod  (rd.) 

320    rods =1  mile  (mi.) 

,760    yards =  1  mile 


,280    feet 


=  1  mile 


6    feet =  1  fathom  (used  in  measur- 
ing the  depth  of  water) 

Square  Measure 


144  square  inches  (sq.  in.) 

9  square  feet 

301  square  yards 

160  square  rods 

640  acres 

640  acres 

100  square  feet 


=  1  square  foot  (sq.  ft.) 

=  1  square  yard  (sq.  yd.) 

=  1  square  rod  (sq.  rd.) 

=  1  acre  (A.) 

=  1  square  mile  (sq.  mi.) 

=  1  section  (sec.) 

=  1  square  (sq.) 


TABLES  OF  WEIGHTS  AND  MEASURES 


Cubic  Measure. 


1,728  cubic  inches  (cu.  in.) . . .  =  1  cubic  foot  (cu.  ft.) 

27  cubic  feet =  1  cubic  yard  (cu.  yd.) 

128  cubic  feet =1  cord  (cd.) 

1  gallon  contains  231  cubic  inches. 

1  bushel  contains  2,150.42  cubic  inches  or  U  cu.  ft, 

(nearly) . 
1  cubic  foot  of  water  contains  7i  gallons  and  weighs 
62i  pounds. 


Liquid  Measure 


1  PINT      I  QUART       I  GALLON 


4    gills  (gi.) =  1  pint  (pt.) 

2    pints =  1  quart  (qt.) 

4    quarts =1  gallon  (gal.) 

3U  gallons =  1  barrel  (bbl.) 

2    barrels =  1  hogshead  (hhd.) 


TABLES  OF  WEIGHTS  AND  MEASURES 


Time  Measure 


60  seconds  (sec.) =1  minute  (min.) 

60  minutes =1  hour  (hr.) 

24  hours. =  1  day  (da.) 

7  days =  1  week  (wk.) 

28,  29,  30,  31  days =  1  month  (mo.) 

12  months =  1  year  (yr.) 

365  days =  1  common  year 

366  days =  1  leap  year 

100  years =1  century 


Table  Used  in  Counting  Merchandise 


12  things =  1  dozen  (doz.) 

12  dozen =1  gross  (gr.) 

12  gross =  1  great  gross  (gt.  gr.) 


ARITHMETIC 

PART  VII 
ADVANCED  LESSONS 


ADVANCED  LESSONS 
PART  VII 

DENOMINATE  NUMBERS 

Lesson  1 
Reduction 

A  ^ ^denominate  number"  is  a  number  which  is  used 
with  the  name  of  a  measure,  as  7  yd.;  5  hours;  $2.; 
40  min. 

Look  at  the  door.  Name  a  denominate  number. 
Name  three  more. 

A  denominate  number  of  only  one  denomination,  as 
8  yd.,  is  a  '^ simple  denominate  number." 

Look  at  the  window  pane.  Name  a  simple  denom- 
inate number. 

A  denominate  number  of  more  than  one  denomina- 
tion,as  2  yd.  5  ft.,  is  a^^compound  denominate  number." 

Look  at  the  clock  and  name  such  a  number.  Find 
another  in  the  room. 

Changing  the  denomination  of  a  number  without 
changing  its  value  is  called  ''reduction." 

6  ft.  =  ?  in.;  2  yd.  =  ?  ft.;  24  in.  =  ?  ft.;  36  in. 
=  ?  yd. ;  1  ft.  =  ?  part  of  a  yd. ; 

Exercise  1 — Oral. 

Reduce : 

L     4    yd.  to  ft.;  3.     7  ft.  to  in.; 

2.     2i  lb.  to  oz.;  4.     2  hr.  to  min.; 


.ARITHMETIC 

5.  3  mo.  to  da.;  11.     Gin.  toft.; 

6.  72  in.  to  yd.;  12.     8  oz.  to  lb.; 

7.  9  yd.  toft.;  13.     3  pk.  tobu.; 

8.  48  oz.  to  lb.;  14.     7pt.  toqt.; 

9.  120  sec.  to  min.;  15.  80  min.  to  hr. ; 
10.     72  in.  to  ft.;  16.  10  hr.  to  min. 


/ 

EXAMPLE:     Reduce  4  yd.  2  ft.  6  in.  to  inches. 

3  ft. 

X4  =  12  ft.; 

Steps 

12  ft 

+  2ft.  =  14  ft.; 

(1)  Reduce  yards  to  feet  and 

12  in 

.  X  14  =  168  in.; 

add  feet,  if  any. 

168  in.  +  6  in.  =  174  in.,  Ans. 

(2)  Reduce    feet    to    inches 

and  add  inches,  if  any. 

Prove  examples  like  this  by  approximation:                       | 

4  yd. 

2  ft.  6  in.  =  almost  5  yd.; 

5  yd.  =  (36"  X  5)  180  in. 

The  answer  is  6 

in.  less,  or  174  in. 

To  reduce  a  compound  denominate  number  to  a 
smaller  denomination,  begin  with  the  largest  denomina- 
tion and  reduce  it  to  the  next  smaller  denomination, 
then  to  the  next  smaller,  and  so  on. 


EXAMPLE:     Reduce  6  da 

.  7  min.  to  minutes. 

Steps 

24  hr.  X  6  =  144  hr.; 

(1)  Reduce  da.   to  hr.  and 

60  min.  X  144  =  8,640  min; 

add  hr.,  if  any. 

8,640  min.  +  7  min.  = 

(2)  Reduce  hr.  to  min.  and 

8,647  min.,  Ans. 

add  min.,  if  any. 

When  a  denomination  is  skipped  reduce  one  step  at 

a  time  as  before,  using  the  skipped  one  in  its  proper 

turn. 

(VII-2) 


DENOMINATE  NUMBERS 

Exercise  2 — Written. 

Reduce: 

1.  5  yd.  2  ft.  to  ft.;  7.  4  lb.  6  oz.  to  oz.; 

2.  5  yd.  2  ft.  8  in.  to  in.;  8.  7  ft.  8  in.  to  in.; 

3.  3  qt.  1  pt.  to  pt. ;  9.  8  yr.  2  mo.  to  mo. ; 

4.  4  bu.  3  pk.  5  qt.  to  qt. ;  10.  1  bu.  2  pk.  2  qt.  to  qt. ; 

5.  3  pk.  1  pt.  to  pt.;  11.  2  yd.  1  ft.  Gin.  to  in.; 

6.  5  da.  6  hr.  20  min.  to  min. ;  12.  3  gal.  2  qt.  to  qt. 


EXAMPLE:     Reduce  174  in.  to  larger  denominations. 

Steps 

(1)  Reduce  in.  to  ft.  saving 
12)  174  (no.  of  in.)  remainder,  if  any. 

3)  14  (no.  of  ft.)  +  6  in.  (2)  Reduce  ft.  to  yd.  saving 

4  (no.  of  yd.)  +  2  ft.  remainder,  if  any. 

4  yd.  2  ft.  6  in.,  Ans.  (3)  Write  last  quotient  plus 

all  remainders. 


To  reduce  a  simple  denominate  number  to  larger 
denominations,  reduce  to  the  next  larger  denomination, 
then  to  the  next  larger, and  so  on,  saving  all  the  remain- 
ders. The  last  quotient  plus  all  the  remainders  is  the 
answer. 

Exercise  3 — Written. 

Reduce : 

1.  12,600  sec.  to  hr.  and  min.; 

2.  6,500  min.  to  largest  denominations; 

3.  762  in.  to  largest  denominations; 

4.  51  pt.  (liquid)  to  largest  denominations; 

5.  51  pt.  (dry)  to  largest  denominations; 

6.  202  qt.  to  largest  denominations; 

(VII-3) 


ARITHMETIC 

7.  195  min.  to  largest  denominations; 

8.  8,647  min.  to  largest  denominations; 

9.  246  oz.  to  largest  denominations; 

10.  8,966  sq.  in.  to  largest  denominations. 

Lesson  2 
Reduction  of  Fractional  Denominate  Numbers 


EXAMPLE:     Reduce  i  yd.  to  inches, 
i  yd.  =  I  of  3  ft.,  which  is  f  or  ^  ft.; 


_  1 


2  of  12  in.,  or  6  in.,  Ans. 
Also:  I  yd.  =  |  of  36  in.,  or  6  in.,  Ans, 


Reduce  the  fraction  of  the  given  denomination  to  the 
next  smaller  denomination;  then  to  the  next  smaller 
again,  until  the  required  denomination  is  reached. 


se  't — wrai. 

1.  ida.  =  ?hr.; 

5.  i  gal. 

=  ?pt.; 

2.  i  hr.   =  ?  min. ; 

6.  i  bu. 

=  ?qt.; 

3.  f  bu.  =  ?pk.; 

7.  f  lb. 

=  ?  oz.; 

4.  ipk.  =  ?qt.; 

8.  i  da. 

=  ?hr. 

9.  Can  days  be  changed  to  hours? 

10.  Can  i  da.  be  changed  to  hours? 

11.  Can  bushels  be  changed  to  pecks? 

12.  Can  T  bu.  be  changed  to  pecks? 

13.  Can  3  pk.  be  changed  to  a  bu.? 

14.  How  many  pk.  make  a  bu.? 

15.  Will  3  pk.  make  a  part  of  a  bu.? 

16.  What  part  of  a  bu.  will  3  pk.  make? 

(VII-4) 


DENOMINATE   NUMBERS 

When  any  denominate  number  or  part  of  a  denomi- 
nate number  does  not  make  a  whole  unit  of  the  next 
larger  denomination,  find  what  part  it  does  make  and 
use  that  fraction. 


Exercise  5 — Oral. 

1.  6  in.  =  ?  part  of  1  ft.;        6.  Change  15  da.  to  mo.; 

2.  8  oz.  =  ?  part  of  1  lb.;       7.  Change  5  da.  to  mo.; 

3.  2  ft.  =  ?  part  of  1  yd.;       8.  Change  3  pt.  to  gal.; 

4.  6  doz.  =  ?  part  of  1  gr. ;     9.  Change  3  pk.  to  bu. ; 

5.  12  hr.  =  ?  part  of  1  da.;    10.  Change  10  oz.  to  lb. 
11.  Can  you  identify  these  as  quickly  as  a  mailman 

does  his  letters?     Try. 

8  (Say  rapidly  '^8  qt.  =  1  pk.") 


4 
24, 

9; 
2; 

5,280 
366 

5^ 
160 

h 

3; 

30i; 

1,760 
128 

100, 
640, 

1,728; 
60; 

16 
144 

27; 

320; 

7 

2, 

3H; 

2,240 

365; 

63; 

12 

30; 

2,000; 

20. 

Exercise  6 — ^ 

Written. 

Reduce : 

1.  4  ft.  6  in 

to  feet; 

2.  3  pk.  1  q 

3.  3  pk.  1  p 

4.  4  hr.  30  I 

t.  to  pecks; 
t.  to  pecks; 
nin.  to  hours; ' 

(VII-5) 

ARITHMETIC 

5.  4  da.  6  hr.  to  days; 

6.  12  ft.  3  in.  to  feet; 

7.  5  yd.  2  ft.  to  yards; 

8.  6  da.  10  hr.  to  days; 

9.  8  ft.  3  in.  to  feet; 

10.  3  bu.  3  pk.  to  bushels; 

11.  9  in.  to  yards; 

12.  1  pt.  to  pecks; 

13.  5  gal.  1  pt.  to  gallons; 

14.  2  ft.  4  in.  to  yards; 

15.  3  qt.  1  pt.  to  gallons; 

16.  30  min.  45  sec.  to  hours; 

17.  7  sq.  ft.  72  sq.  in.  to  square  yards; 

18.  A  pk.  to  smaller  denominations; 

19.  i  yd.  to  smaller  denominations; 

20.  A  gt.  gr.  to  smaller  denominations. 

Lesson  3 
Addition  of  Denominate  Numbers 


EXAMPLE; 

The  sum  of  the  hour  column  is  31  hr.  which  is 

reduced  to  1  da.  7  hr. ;  the  7  hr.  are  written  in 

wk. 

da. 

hr. 

the  hour  column  and  the  1  da.  is  carried  to  the 

4 

6 

8 

day  column;  the  sum  of  the  day  column  (in- 

5 

3 

12 

cluding  the  1  da.  carried  from  the  hour  column) 

- 

4 

7 

is  14  da.  which  is  reduced  to  2  wk.  0  da.;  the 

15 

- 

4 

0  da.  are  written  in  the  day  column  and  the 

26 

0 

7 

2  wk.  are  carried  to  the  week  column;  the  sum 
of  the   week   column    (including  the  2   wk. 
carried  from  the  day  column)  is  26  wk.,  which 
is  written  \xy  the  week  column. 

(VII-6) 


DENOMINATE   NUMBERS 

In  adding  denominate  numbers,  a  separate  column 
must  be  used  for  each  denomination  to  be  added,  as 
we  must  always  bear  in  mind  that  unlike  numbers, 
quantities  or  things  cannot  be  added. 

After  finding  the  sum  of  the  addends  of  the  smallest 
denomination,  the  sum  must  be  reduced  so  that  all 
units  of  a  larger  denomination  which  are  contained  in 
it  may  be  carried  and  added  to  the  addends  of  the 
larger  denomination. 

Exercise  7 — Written. 
Add  and  prove: 

The  Iceman's  Problems 


1.  To  fill  a  large  refrigerator,  4  loads  of  ice  were 
necessary ;  how  much  did  it  cost  to  fill  this  refrig- 
erator if  ice  is  worth  30^  per  100  lb.  and  the  four 
loads  weighed  as  follows: 

(VII-7) 


ARITHMETIC 

T. 

cwt. 

lb. 

2 

6 

25 

2 

4 

50 

2 

8 

75 

1 

3 

30 

1 

4 

20 

Can  the  iceman  save  time  in  finding  his  total 
dehvery  by  adding  groups?  Try  it.  First  add 
all  the  lb.,  then  all  the  cwt.,  then  the  T.  Now- 
express  the  sum  in  cwt.  because  the  price  is  30^;^ 
per  cwt.     Now  make  out  the  bill. 

2,  If  he  delivered  3  cwt.  50  lb.;   2  cwt.  25  lb.;  and 

6  cwt.  25  lb.;  what  is  the  value  of  the  whole 
delivery  at  25ff  per  cwt.? 

3.  If  he  received  20  T.  on  one  delivery,  17  T.  6  cwt. 

50  lb.  on  another  dehvery,  and  2  T.  3  cwt.  5  lb. 
on  a  third  delivery,   how  many  cwt.   did  he 


receive? 


4.  If  he  packed  14  T.  5  cwt.  in  one  car,  20  T.  6  cwt. 

in  another,  and  12  T.  4  ewt.  in  another,  how 
many  tons  did  he  pack? 

5.  If  he  sold  ice  to  restaurants  at  25^  per  cwt.,  how 

much  did  he  collect  for  the  following  deliveries: 


T. 

cwt. 

lb. 

1 

18 

70 

1 

14 

80 

1 

12 

40 

2 

1 

50 

2 

10 

(VII-8) 


DENOMINATE   NUMBERS 


The  Merchant's  Problems 


6.  What  is  the  cost  of  the  oil  cloth  needed  to  cover 


4  halls,  if  a  square  yard  costs 
of  the  halls  is  as  follows: 

sq.  yd.  sq.  ft. 

6  8 

10  8 

12  7 

4  7 


.44  and  the  area 


7.  A  wholesale  stationer  received  3  orders  for  pencils; 
if  he  sold  the  pencils  for  1^  each  and  the  orders 
were  for  the  following  quantities,  how  much  did 
he  receive  for  all  the  pencils? 

gr.         doz. 
4  6 

8            2 
6 8 

(VII-9) 


ARITHMETIC 

8.  If  one  bolt  of  ribbon  has  6  yd.  2  ft.  left  on  it, 

another  bolt  has  8  yd.  1  ft.,  another  bolt  has  2 
yd.  1  ft.,  how  many  yards  of  ribbon  are  there? 
What  is  the  total  value  at  QO^  a  yard? 

9.  If  1 1  yd.  of  ribbon  is  added  to  a  bolt  containing 

4  yd.  2  ft.  6  in.,  how  many  yards  are  there  in  all? 
10.  If  one  strip  of  carpet  measures  5  yd.  2  ft.  7  in. 
and  another  measures  4  yd.  2  ft.  9  in.,  how  many 
yards  are  there  in  all? 

The  Laborer's  Problems 


11.  A  laborer  working  for  40fZ^  an  hour  would  receive 
how  much  money  for  doing  these  3  pieces  of 
work  if  8  hours  is  considered  a  day's  work: 

da.  hr.  min. 

4  6  45 

8  4  30 

7  3  15 

(VII-IO) 


DENOMINATE   NUMBERS 

12.  A  laborer  working  at  40^  an  hour,  worked  as 
follows  during  three  weeks;    how  much  did  he 
earn  if  9  hours  is  considered  a  day's  work: 
da.         lir.         min. 

4  6  30 

5  7  15 
3            4  — 

Lesson  4 
Subtraction  of  Denominate  Numbers 


EXAMPLE: 

yd.     ft.     in. 

As  10  in.  cannot  be  subtracted  from  6  in.,  1  of  the 

4       1      18 

2  ft.  must  be  changed  to  12  in.  and  added  to  the 

i       t        % 

6  in.,  making  18  in.;    IS  in.  —  10  in.  =  8  in.; 

1       1      10 

1  ft.  -  1  ft.  =  0  ft.;  4  yd.  -  1  yd.  =  3  yd. 

3       0        8 

As  in  addition,  a  separate  column  must  be  used  for 
each  denomination  to  be  subtracted. 

When  the  subtrahend  of  any  denomination  is  larger 
than  the  minuend  of  that  denomination,  1  unit  of  the 
next  larger  denomination  of  the  minuend  must  be 
changed  into  units  of  the  smaller  denomination  and 
added  to  the  other  units  of  that  denomination  before 
subtracting. 


cercise  8 — Written. 

Subtract  and  prove: 

1. 

2. 

hhd.     bbl.    gal. 

mi.    rd. 

yd. 

3          1       18 

5    240 

4 

1          1       20 

(VII-ll) 

2    300 

5 

ARITHMETIC 

3.  4. 

gal.         qt.  pt.  bu.  pk.  qt. 

12          2  —  18       1  4       ' 

5          3         1  6      3  6 

5.  If  a  piece  of  cloth  containing  4  sq.  yd.  2  sq.  ft. 

72  sq.  in.  is  removed  from  a  bolt  containing  25 
sq.  yd.,  how  much  material  remains  on  the  bolt? 
What  is  it  worth  at  $3.60  per  sq.  yd.? 

6.  From  a  box  containing  1  gt.  gr.  of  buttons,  8  gr. 

4|  doz.  are  sold;  how  many  buttons  remain? 
What  are  they  worth  at  36^  per  gross? 

7.  A  number  of  men  were  digging  a  trench  a  mile 

long;  how  many  yards  had  they  still  to  dig 
when  148  rd.  had  been  completed? 

8.  A   train  required   20  hr.   45  min.   to  run  from 

Chicago  to  New  York;  how  long  had  it  still  to 
travel  at  9.30  p.  m.  if  it  left  Chicago  at  12.40  p.  m. 

Lesson  5 
Multiplication  of  Denominate  Numbers 


EXAMPLE:     5  yd.  2  ft.  7  in.  X  6  =  ? 

The  product  of  7  in.  X  6  is  42  in.;  42  in.  is 
reduced  to  3  ft.  6  in.;  write  6  in.  in  the  prod- 
uct and  carry  3  ft.;  the  product  of  2  ft.  X  6 
is  12  ft.;  12  ft.  +3  ft.  (carried)  =  15  ft.; 
15  ft.  is  reduced  to  5  yd.  0  ft. ;  write  0  ft.  in  the 
35        0        6  product  and  carry  5  yd.;   the  product  of  5  yd. 

X  6  is  30  yd.;   30  yd.  +  5  yd.  (carried)  =  35 
yd.;  write  35  yd.  in  the  product. 

(VII-12) 


yd. 

ft. 

in. 

5 

2 

7 
6 

DENOMINATE    NUMBERS 

The  product  of  the  smallest  denomination  must  be 
reduced  so  that  all  units  of  a  larger  denomination  which 
may  be  contained  in  it  can  be  carried  and  added  to  the 
product  found  by  multiplying  the  larger  denomination. 


Exercise  9 — Written. 

Multiply  and  prove: 

1. 

da.      hr.    min. 
12       10      45 
20 


2. 

hhd.    bbl.  gal. 
4       1       7 
9 


3.  What  is  the  cost  of  laying  carpet  in  four  rooms 

at  $3.00  per  square  yard,  if  the  area  of  each 
room  is  20  sq.  yd.  4  sq.  ft.  18  sq.  in.?  (Sugges- 
tion: Find  total  area  first.) 

4.  If  the  circumference  of  a  wagon  wheel  is  3  yd. 

1  ft.  6  in.,  how  far  has  the  w^agon  traveled  when 
the  w^heel  has  turned  1,000  times? 


(VII-13) 


ARITHMETIC 

5.  What  is  the  total  length  of  9  pieces  of  rope,  if 

each  piece  measures  9  rd.  3  yd.  2  ft.?     What  is 
the  rope  worth  @  2^  per  yard? 

6.  Find  the  volume  of  40  blocks  of  granite,  the  vol- 

ume of  each  block  being  1  cu.  yd.  24  cu.  ft.  144 
cu.  in. 

Lesson  6 
Division  of  Denominate  Numbers 


EXAMPLE:     (Short  Division)  5  hr.  40  min.  40  sec.  -^  2  =  ? 

5  hr.  -^  2  =  2  hr.  and  1  hr.  remaining  to  be 

hr.     min.     sec.  changed  to  minutes  which  are  added  to 

2       50        20  the  other  40  min .  of  the  dividend ;  60  +  40 

2)  5       40        40  =  100    min.;     100    min.  H-  2  =  50    min. 

and  no  minutes  remaining  to  be  changed 

to  sec;  40  sec.  -^  2  =  20  sec. 

EXAMPLE:     (Long  Division)  500  A.  83  sq.  rd.  8  sq.  yd.  ^  45  =  ? 

A.  sq.  rd.   sq.  yd. 
11         19         19 
45)  500        83  8 

45 
50  500  A.  -f-  45  =  11  A.  and  5  A.  remaining 

45  to  be  changed  into  800  sq.  rd.  which 

Rem.  5  =  800  are'added  to  the  83  sq.  rd.,  making  883 

883  sq.  rd.;   883  sq.  rd.  h-  45  =  19  sq.  rd. 

45  and  28  sq.  rd.  remaining  to  be  changed 

43§  into  847  sq.  yd.  which  are  added  to  the 

405  8  sq.  yd.,  making  855  sq.  yd.;    855 

Rem.~28  =847         sq.  yd.  -=-  45  =  19  sq.  yd. 
855 

405 
405 


(VII-14) 


DENOMINATE   NUMBERS 

In  dividing  to  find  one  of  the  equal  parts  of  a  com- 
pound denominate  number,  we  divide  the  largest 
denomination  first,  and  any  remainder  from  this  denom- 
ination is  changed  into  units  of  the  next  smaller  denom- 
ination and  these  are  added  to  the  other  units  of  that 
denomination  before  dividing  it.  The  remainder  from 
the  smallest  denomination  is  written  in  the  form  of  a 
fraction  as  in  ordinarv  division. 


EXAMPLE: 

How  many  ribbon  bows  can  be  made  from  10  yd. 
2  ft.  6  in.  of  ribbon,  if  2  ft.  6  in.  are  used  for 
each  bow? 

10  yd 

2  ft.  6  in.  = 
2  ft.  6  in.  = 

390  in.; 
30  in.; 

390 

in.  ^  30  in. 

=  13  (number  of  bows) 

Ans. 

In  dividing  to  find  how  many  times  one  compound 
denominate  number  is  contained  in  another  compound 
denominate  number,  we  first  reduce  both  compound 
denominate  numbers  to  simple  denominate  numbers  of 
the  same  denomination,  then  we  divide  in  the  usual 
way. 

Exercise  10 — Oral. 

A  test  for  you.     Study  to  answer  these  promptly  and 
in  good  English. 

1.  In  adding  and  subtracting  denominate  numbers, 

why  do  we  use  a  separate   column  for  each 
denomination? 

2.  In  adding  denominate  numbers,  what  is  done  with 

the  sum  of  the  first  column  before  finding  the 
sum  of  the  next  column? 
(VII-15) 


ARITHMETIC 

3.  Explain  just  what  you  would  do  to  subtract  13 

cwt.  from  1  T.  2  cwt. 

4.  Tell  how  to  multiply  4  yd.  2  ft.  6  in.  by  10. 

5.  Tell  how  to  reduce  2,400  ft.  to  rods. 

6.  Tell  how  to  divide  12  lb.  8  oz.  by  10. 

7.  How  do  we  change  inches  to  feet?    Feet  tc  yards? 

Yards  to  rods?     Rods  to  miles? 

8.  How  do  we  change  square  inches  to  square  feet? 

Square  feet  to  square  yards?  Square  yards  to 
square  rods?  Square  rods  to  acres?  Acres  to 
square  miles? 

9.  How  do  we  change  cubic  yards  to  cubic  feet? 

Cubic  feet  to  cubic  inches? 

10.  How  do  we  change  hogsheads  to  barrels?     Bar- 

rels to  gallons?  Gallons  to  quarts?  Quarts  to 
pints?     Pints  to  gills? 

11.  How  would  you  find  how  many  times  1  hr.  20  min. 

is  contained  in  5  hr.  20jxiin.? 

Exercise  11 — Written. 

Solve  and  prove : 

1.  263  da.  23  hr.  30  min.  ^  6  =  ? 

2.  150  mi.  6  rd.  3  yd.  -^  12  =  ? 

3.  406  cu.  yd.  23  cu.  ft.  50  cu.  in.  -^  10  =  ? 

4.  10  sq.  yd.  8  sq.  ft.  96  sq.  in.  -^  48  =  ? 

5.  341  T.  -T-  40  =  ? 

6.  41  hhd.  1  bbl.  -f-  9  =  ? 

7.  A  trench  1  mile  long  is  divided  into  10  equal 

sections;  how  many  yards  long  is  each  section? 

8.  The  perimeter  or  distance  around  a  square  is  19 

yd.  1  ft. ;  what  is  the  length  of  one  of  the  sides? 
What  is  the  area  in  sq.  yd.? 
(VII-16) 


DENOMINATE   NUMBERS 

9.  What  is  the  average  weight  of  each  of  the  following 
5  machines: 

The  first  weighs  1  T.  4  cwt.  78  lb.; 
The  second  weighs  9  cwt.  50  lb.; 
The  third  weighs  2  T. ; 
The  fourth  weighs  2  T.  3  cwt. ; 
The  fifth  weighs  2  T.  1  cwt.  22  lb. 

Exercise  12 — Oral. 

1.  What  table  of  measures  is  used  for  measuring  dis- 

tance? Height?  Length?  Be  ready  to  write 
it  on  the  board  quickly. 

2.  What  table  of  measures  is  used  for  weighing  all 

common  articles  such  as  coal,  meat,  sugar,  etc.? 
Say  this  table. 

3.  What  table  of  measures  is  used  for  measuring 

hquids  such  as  water,  milk,  oil,  etc.?  Be  ready 
to  say  the  table. 

4.  What  table  of  measures  is  used  for  measuring  the 

quantity  of  vegetables  and  grains  such  as  oats, 
wheat,  potatoes,  etc.?    Be  ready  to  write  it. 

5.  What  table  of  measures  is  used  for  counting  articles 

such  as  buttons,  eggs,  etc.?  Write  all  the  num- 
bers and  let  your  classmates  identify  them.  Be 
ready. 

6.  What  table  of  measures  is  used  for  measuring 

time?     Say  this  table. 

7.  Name  the  table  used  for  measuring  the  area  of 

surfaces.     Do  you  remember  this  table? 

8.  What  table  of  measures  is  used  for  measuring  the 

volume  of  solids?     Be  ready  to  WTite  it. 
(VII-17) 


ARITHMETIC 

9.  How  many  and  what  dimensions  has  a  line?     A 
surface?     A  solid? 

10.  What  is  the  size  of  a  board  foot?     What  is  the 

volume  of  a  board  foot? 

11.  Draw  a  board  foot;  a  cubic  foot;  a  square  foot; 

a  foot. 

12.  Can  you  identify  each  quickly? 

Exercise  13 — Written. 

1.  How  many  pieces  of  tile  6^'  X  6''  will  be  needed  to 

cover  a  floor  12'  X  187  Can  you  use  cancella- 
tion here?    Try  12  X  18  4-  i  -r-  i. 

2.  In  a  certain  orchard  there  are  400  trees  planted 

in  20  rows  of  20  trees  each;  if  the  area  of  this 
orchard  is  10  A.,  how  much  area  is  allowed  for 
each  tree? 

3.  If  3i  yd.  of  brass  tubing  weigh  1  lb.,  how  many 

feet  of  tubing  weigh  1  oz.? 

4.  How  many  36  ft.  lengths  of  wire  are  needed  to 

build  a  wire  fence  6  wires  high  around  a  field 
16  rd.  X  10  rd.? 

5.  How  many  times  can  a  2i  gal.  pail  be  filled  from 

a  tank  containing  600  pints? 

6.  What  is  the  value  of  4  gal.  3  qt.  1  pt.  of  gasolene 

@  24^  per  gallon? 

7.  What  decimal  fraction  of  a  great  gross  is  equal 

to  1  gr.  6  doz.? 

8.  At  $5.00  per  dozen,  what  is  the  cost  of  4  gt.  gr. 

5  gr.  of  books? 

9.  A  grocer  bought  50  bu.  potatoes  @  GOff  per  bu., 

and  sold  i  of  them  @  20^  per  pk.  and  the  other 
(VII-18) 


DENOMINATE   NUMBERS 

half  @  25p  per  pk. ;  what  was  the  grocer's  profit? 
What  was  the  per  cent  of  profit? 

10.  A  grocer  invests  $100.00  in  apples  which  he  sells 

at  28%  profit;  if  he  sells  the  apples  at  32^  per 
pk.,  how  many  bushels  did  he  buy?  What  did 
he  pay  for  each  bushel? 

11.  A  family  using  12  T.  of  coal  during  the  winter 

bought  17,000  lb.  @  $5.00  per  T.  and  the 
balance  @  $6.00  per  T. ;  what  was  the  total  cost 
of  the  coal?  How  much  could  have  been  saved 
by  bu}dng  the  entire  supply  at  $5.25  per  T.? 

12.  If  an  acre  planted  in  hay  yields  H  T.  when  hay 

is  worth  $12.00  per  T.,  while  an  acre  planted  in 
oats  yields  30  bu.  when  oats  is  worth  50^  per 
bu.,  is  it  more  profitable  to  raise  hay  or  oats? 
If  you  had  a  square  mile  of  land  what  could  you 
gain  by  raising  the  more  profitable  crop? 

13.  If  a  printing  press  can  print   7,500  completed 

magazines  in  1  hr.  of  continuous  printing,  how 
many  magazines  can  it  print  in  6  w^k.  of  48  hr. 
each,  allowing  1  hr.  out  of  each  8  hr.  day  for 
necessary  delays  on  account  of  oiling,  changing 
paper,  etc.? 

14.  If  it  takes  a  man  1  hr.  10  min.  to  make  a  certain 

tool,  how  many  such  tools  could  he  make  in 
46  hr.  40  min.  of  work?     Compare  carefully. 

15.  A  cubic  foot  of  water  weighs  62.5  lb.;  what  is  the 

weight  of  the  water  which  would  completely  fill 
a  tank  6  ft.  long,  2  ft.  wide,  and  18  in.  deep? 

16.  How  many  1  lb.  8  oz.  packages  of  sugar  can  be 

filled  from  a  barrel  containing  4  cwt.  20  lb.? 
(VII-19) 


ARITHMETIC 


17.  In  a  bed  of  pansies,  each  plant  is  given  36  sq.  in. 

of  ground  space;  how  many  plants  can  be 
planted  in  5  sq.  yd.  8  sq.  ft.  108  sq.  in.? 

18.  How  many  jars  of  jam  each  containing  2  qt.  1  pt. 

can  be  filled  from  a  pail  containing  20  gal.? 

19.  A  carpenter  laying  flooring  requires  4  hr.  15  min. 

to  lay  6  sq.  yd.  6  sq.  ft.;  how  long  will  it  take 
him  to  lay  the  floor  in  a  room  4  yd.  long  and 
3  yd.  1  ft.  wide? 

20.  A  carpenter  laying  flooring  requires  4  hr.  15  min. 

to  lay  6  sq.  yd.  6  sq.  ft.;  how  much  flooring 
can  he  lay  in  25  hr.  30  min.? 

Lesson  7 


Measuring  Land 


N 


N 


W 


6 

5 

A- 

3 

2 

1 

7 

8 

3 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22. 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

E  W 


N.W. 

Qtr. 
(I60    A) 


Qtr. 


OF 


OF 
N.E.;4: 


OF 


OF 


S.  EL 

Qtr 


(160    A  )         (160.  A 


s 

(Figure  1) 

1  Township. 

36  Sections. 

36  Square  Miles. 


a 

(Figure  2) 

1  Section. 

1  Square  Mile. 

640  Acres. 


Figure  2  shows  a  section  divided  into  4  quarters,  the  N.  E.  quarter 
of  the  section  being  divided  into  4  quarters  of  40  A.  each. 

The  greater  part  of  the  land  in  the  United  States  is 
divided  into  townships. 

(VII-20) 


DENOMINATE   NUMBERS 

A  township  is  a  piece  of  land  6  miles  square ;  that  is, 
it  is  6  miles  long  and  6  miles  wide;  hence,  a  township 
contains  36  square  miles  or  '^sections/'  a  ^'section" 
being  1  square  mile.  These  36  sections  are  numbered 
from  1  to  36  as  shown  in  the  illustration.     Count  them. 

Each  section  is  divided  into  four  quarters  and  each 
of  these  four  quarters  is  divided  into  four  quarters; 
hence,  a  section  will  have  a  North  East  (N.  E.)  quarter, 
a  North  West  (N.  W.)  quarter,  a  South  East  (S.  E.) 
quarter,  and  a  South  West  (S.  W.)  quarter,  and  each 
of  these  quarters  will  have  a  N.  E.,  N.  W.,  S.  E.,  and 
S.  W.  quarter.     N.  E.  i;  S.  E.  i;   N.  W.  i;  S.  W.  h 

As  a  square  mile  or  section  contains  640  acres,  each 
quarter-section  contains  160  acres,,  and  each  quarter  of 
a  quarter-section  contains  40  acres. 

Exercise  14 — Oral  and  Written. 

1.  Draw  the  plan  of  a  township. 

2.  How  long  and  how  wide  is  a  township? 

3.  How  many  square  miles  are  there  in  a  township? 

4.  What  is  each  square  mile  in  a  township  called? 

5.  Say  how  the  sections  in  a  township  are  numbered. 

6.  Number  the  sections  on  your  plan. 

7.  Draw  the  plan  of  a  section. 

8.  What  is  the  area  of  a  section  in  square  miles? 

In  acres? 

9.  Show  the  N.  E.  i;  N.  W.  i;  S.  E.  i;  S.  W.  h 

10.  How  many  acres  are  there  in  each  of  these  quarters? 

11.  Show  a  160-acre  farm  on  your  plan.     Locate  it 

as  a  part  of  the  section. 

12.  Show  the  East  i  of  N.  E.  h 

(VII-21) 


ARITHMETIC 

13.  Show  a  40-acre  farm  on  your  plan.     How  do  you 

write  its  location? 

14.  How  many  acres  are  there  in  a  quarter  of  a  quar- 

ter-section? 

15.  Show  a  320-acre  farm  on  your  plan.     Read  its 

location. 

16.  How  many  acres  are  there  in  the  N.  W.  i  of 

N.  E.  i?     (See  Figure  2,  Page  20.) 

17.  How  many  acres  are  there  in  the  S.  E.  i  of  N.  E. 

i?     (See  Figure  2,  Page  20.) 

Exercise  15 — Written. 

1.  A  820-A.  farm  is  what  part  of  a  section? 

2.  A  160-A.  farm  is  what  part  of  a  section? 

3.  An  80- A.  farm  is  what  part  of  a  section?     What 

part  of  a  quarter-section? 

4.  A  40-A.  farm  is  what  part  of  a  section?     What 

part  of  a  quarter-section? 

5.  What  are  the  dimensions  of  a    half-section  in 

miles?     In  rods? 

6.  What  are  the  dimensions  of  a  quarter-section  in 

miles?     In  rods? 

7.  Wliat  are  the  dimensions  of  half  of  a  quarter- 

section  in  miles?     In  rods? 

8.  What  are  the  dimensions  of  quarter  of  a  quarter- 

section  in  miles?     In  rods? 

9.  Draw  a  plan  of  a  section  and  shade  the  South 

half  of  the  N.  E.  quarter  with  horizontal  lines. 

10.  If  the  drawing  shows  a  quarter  of  a  quarter-sec- 

tion, how  many  acres  are  planted  in  corn? 

11.  How  many  acres  are  planted  in  oats? 

(YII-22) 


DENOMINATE   NUMBERS 


N.E.>^  OF  S.W.;^  OF    SEC.  lO 

CORN 

OMTS 

WHEAT 

POTATOES 

OATS 

WHEAT 

CORN 

RYE. 

BARLEY 

A  Quarter  of  a  Quarter-Section 

12.  How  many  acres  are  planted  in  wheat? 

13.  How  many  acres  are  planted  in  potatoes? 

14.  How  many  acres  are  planted  in  rye? 

15.  How  many  acres  are  planted  in  barley? 

16.  How  many  rods  of  fence  are  required  for  one  of 

the  com  fields?     How  many  rods  for  the  fence 
between  wheat  and  potatoes? 

17.  How  many  miles  of  fence  are  required  to  go  around 

the  N.  E.  i  of  S.  W.  i? 

18.  How  man}^  rods  of  fence  are  required  on  the  south 

side  of  the  rye  field? 

19.  How  many  rods  of  fence  are  required  on  the  north 

side  of  the  entire  Section  10? 

20.  What  is  the  perimeter  of  the  entire  Section  10  in 

miles? 

21.  What  is  the  perimeter  of  the  rye  field? 

22.  What   is   the   total   perimeter   of   the   two   com 

fields?     (Answer  in  rods.) 

(VII-23) 


ARITHMETIC 


Lesson  8 

Paper  Measure 

24  sheets  =  1  quire  (qr.) 
20  quires  =  1  ream  (rm.) 
500  sheets  =  1  printer's  ream. 


EXAMPLE:     Find  the  cost  of  a  ream  of  34"  X  44"  bond  paper 
(a  10?^  per  lb.,  if  a  ream  of  17"  X  22"  =  16#. 

1,496  sq.  in.  is  a  larger  area  than  374 
34"  X  44"  =  1,496  sq.  in.  sq.  in.;  therefore,  the  ratio  of  the 
17"  X  22"  =     374sq.  in.        weights     must      be      1,496:374  = 


1    4  9  6  or  4.  • 

TT^  or  '±, 


1  ream  of  34"  X  44"  paper  therefore  weighs  16#  X  4,  or  64#,  and 
@  10?^  per  lb.  costs  $6.40,  Ans. 


Writing  paper  and  fine  stationery  is  sold  by  the  quire 
consisting  of  24  sheets,  or  by  the  ream  consisting  of 
480  sheets. 

Paper  used  for  printing  purposes  is  sold  by  the  pound, 
on  the  basis  of  a  certain  weight  for  a  ream  of  500  sheets ; 

(VII-24) 


DENOMINATE   NUMBERS 

thus,  if  a  ream  of  paper  of  a  certain  size  weighs  16#, 
and  the  price  of  the  paper  is  lOj^  per  pound,  the  ream 
would  cost  10^  X  16,  or  $1.60;  250  sheets  or  i  ream 
would  weigh  i  of  16#,  or  8#  and  would  cost  80j^;  3 
reams  would  weigh  16#  X  3,  or  48#  and  would  cost 
$4.80. 

The  basis  of  weight  naturally  varies  with  the  size  of 
the  sheet;  hence,  if  we  know  the  basis  of  weight  for  a 
ream  of  paper  of  any  size,  we  can  easily  find  the  weight 
of  a  ream  of  any  other  size  by  comparing  the  two  areas. 


EXAMPLE:     Find  the  cost  of  3,000  sheets  of  22"  X  34"  paper 

@  8^  per  lb.  on  the  basis  of  a  ream  of  17"  X  22" 

=  20#.    • 

34"  paper  will  weigh  more  than  17" 

paper;   therefore,  the  ratio 

is  34: 17  =  ff  or  2. 

Here  the  dimension  22"  is 

1  ream  of  22"  X  34"  paper  therefore 

common  to  both  sizes  of 

weighs  20#  X  2,  or  40#; 

paper;    therefore,  we  use 

3,000  sheets  =  6  rm.;    40#  X  6  = 

the  ratio  of  the  two  unlike 

240#; 

dimensions  only,  instead 

240#  %Si  =  $19.20,  Ans. 

of  the  ratio  of  the  two 

areas. 

When  the  two  sizes  of  paper  have  one  dimension  the 
same,  this  dimension  can  cause  no  change  in  weight  or 
money;  therefore,  find  the  ratio  of  the  unUke  dimen- 
sions only,  as  this  ratio  is  the  same  as  the  ratio  of  the 
two  areas. 

Exercise  16 — Oral. 

1.  Say  the  table  of  Paper  Measure. 

2.  If  fine  writing  paper  costs  20^  per  quire,  what  is 

the  cost  of  1  ream? 

(MI-25) 


ARITHMETIC 

3.  If  linen  writing  paper  cost  1^  per  sheet,  what  is 

the  cost  of  1  ream? 

4.  How  many  sheets  are  there  in  a  printer's  ream? 

5.  If  a  ream  of  a  certain  kind  of  paper  weighs  20# 

and  the  price  is  8^  per  lb.,  what  is  the  cost  of 
the  entire  ream? 

6.  If  a  ream  of  a  certain  kind  of  paper  weighs  30#, 

what  is  the  weight  of  7  reams  of  the  same  paper? 

7.  If  a  ream  of  a  certain  size  of  paper  weighs  40#, 

what  is  the  weight  of  a  ream  of  the  same  kind 
of  paper  when  the  sheets  are  double  the  size? 

8.  If  you  know  the  weight  of  a  ream  of  a  certain 

kind  of  paper  of  a  certain  size,  how  can  you 
find  the  weight  of  a  ream  of  the  same  paper  of 
any  other  size?  Is  there  such  a  problem  in 
Exercise  17? 

9.  When  one  dimension  is  common  to  two  sizes  of 

paper,  how  do  you  find  the  ratio  quickly?     Can 
you  find  such  a  problem  in  Exercise  17? 
10.  What  kind  of  paper  is  sold  480  sheets  to  the  ream? 
What  kind  is  sold  500  sheets  to  the  ream? 

Exercise  17 — Oral  and  Written. 

A.  Tell  how  to  do  these  examples.     Point  out  the 
comparison  in  each  example. 

1.  Find  the  cost  of  750  sheets  of  bond  paper  @  12j!^ 

per  lb.  on  the  basis  of  1  rm.  22"  X  34''  =  40#. 

2.  Find  the  weight  of  1  ream  of  36"  X  48"  ledger 

paper  on  the  basis  of  1  rm.  24"  X  36"  =  44#. 
What  difference  do  you  notice  in  the  size  of  the 
sheets? 

(VII-26) 


DENOMINATE   NUMBERS 

3.  Find  the  weight  of  8  rm.  17"  X  22"  flat  paper  on 

the  basis  of  1  rm.  22"  X  34"  =  36#.     Compare 
the  weights  of  1  ream  of  each  size. 

4.  Find  the  cost  of  1  rm.  18"  X  24"  Unen-finish  paper 

@  9^  per  lb.  on  the  basis  of  1  rm.  36"  X  48" 
=  72#. 

5.  Find  the  cost  of  20  rm.  42"  X  42"  super-coated 

paper  @  7^  per  lb.  on  the  basis  of  1  rm.  42"  X 
56"  =  212#. 

6.  Find  the  cost  of  6,000  sheets  of  17"  X  22"  machine- 

finish  paper  @  6t^  per  lb.  on  the  basis  of  1  rm. 
22"  X  34"  =  32#. 

7.  Find  the  cost  of  4  tons  of  10"  X  30"  manila  wrap- 

pers @  5^  per  lb.  on  the  basis  of  1  rm.  30"  X 
40"  =  36#. 

8.  Find  the  cost  of  4,000  white   manila  10"  X  30" 

wrappers  @  $5.00  per  cwt.  on  the  basis  of  1  rm. 
30"  X  40"  =  36#. 

9.  Find  the  cost  of  10  rm.  35"  X  40"  brown  wrappers 

@  $120.00  per  T.  on  the  basis  of  24"  X  35"  = 
30#  to  the  ream. 
10.  Find    the    cost    of    10,000    sheets    of   42"  X  56" 
machine-finish  paper  @  $5.70  per  cwt.  on  the 
basis  of  1  rm.  21"  X  28"  =  50#. 

B.  Now  work  all  of  the  examples. 

Lesson  9 

Printers'  Type  Measure 

12  points    =  1  pica 
6  picas  (72  points)  =  1  inch 
(VII-27) 


ARITHMETIC 


In  reading  a  newspaper,  magazine,  or  book  (this 
arithmetic,  for  example)  have  you  noticed  the  perfect 
uniformity  of  all  the  characters  in  each  style  of  type 
used,  and  the  uniformity  of  the  width  of  the  columns 
and  pages?  If  you  haven't  noticed  these  things,  it  is 
on  account  of  the  very  uniformity  which  exists,  for, 
if  even  a  single  letter  in  a  line  were  iho  of  an  inch 
larger  or  smaller  than  the  others,  you  would  quickly 
notice  that  something  was  wrong. 

To  obtain  this  uniformity,  printers  require  units  of 
measurement  much  smaller  than  the  inch  or  even  16th's 
or  32d's  of  an  inch;  so  they  have  divided  the  inch  into 
72d's  (approximately)  and  they  call  t2  of  an  inch  a 
''point,"  and  12  of  these  points  are  called  a  ''pica,'' 
so  there  are  6  picas  to  an  inch. 

The  height  of  type  is  usually  measured  by  points, 
thus:  as  there  are  72  points  to  an  inch,  6-point  type  is 
T2  of  an  inch  high;  8-point  type  is  i  of  an  inch  high; 
12-point  t3^e  is  i  of  an  inch  high;  etc. 

The  length  and  width  of  the  surface  covered  by  the 
printing  on  a  page  is  usually  measured  by  picas,  thus: 

(VII-28) 


DENOMINATE   NUMBERS 

as  there  are  6  picas  to  an  inch,  a  page  of  type  4  inches 
wide  is  24  picas  wide;  a  page  of  type  3|  inches  wide  is 
21  picas  wide;  a  page  of  type  lOJ  inches  long  is  63 
picas  long;  etc. 

The  area  of  the  surface  covered  by  the  printing  on  a 
page  is  usually  measured  by  a  unit  called  an  ''em." 
This  unit  is  a  square  whose  face  is  as  many  points  high 
and  wide  as  there  are  points  in  the  size  of  type  being 
used  on  a  particular  piece  of  printing.  Thus,  if  6- 
point  type  is  being  used,  an  ''em"  is  6  points  high  and 
6  points  wide;  therefore,  there  would  be  12  six-point 
"ems"  (72  -^  6)  in  a  line  1  inch  long,  and  12  six-point 
"ems"  in  a  column  1  inch  long  and  1  "em"  wide;  in 
other  words,  there  are  144  six-point  "ems"  to  the 
square  inch.  Eight-point  type  runs  9  "ems"  to  the 
inch  (72  -r-  8)  or  81  "ems"  to  the  square  inch;  etc. 

This  paragraph  is  set  in  12-point  type,  21  picas 
wide;  this  gives  us  6  lines  to  the  inch,  because 
there  are  6  times  12  points  in  72  points.  Since  a 
12-point  em  is  12  points  square,  there  are  6  ems 
to  the  inch,  or  36  ems  to  the  square  inch ;  there- 
fore, this  paragraph  is  21  ems  wide  and  7  ems 
deep  and  contains  147  ems. 

This  paragraph  is  set  in  8-point  tvpe,  16  picas 
wide;  this  gives  us  9  Knes  to  the  inch,  because 
there  are  9  times  8  points  in  72  points.  Since  an 
8-point  em  is  8  points  square,  there  are  9  ems  to  ^ 
the  inch,  or  81  ems  to  the  square  inch;  therefore, 
this  paragraph  is  24  ems  wide  and  7  ems  deep,  and 
contains  168  ems. 

This  paragraph  is  set  in  6-point  type,  12 
picas  wide;  this  gives  us  12  lines  to  the  inch, 
because  there  are  12  times  6  points  in  72 
points.  Since  a  6-point  em  is  6  points  square, 
there  are  12  ems  to  the  inch,  or  144  to  the 
square  inch;  therefore,  this  paragraph  is  24 
eais  wide  and  7  ems  deep,  and  contains  168  ems. 

(YII-29) 


ARITHMETIC 

Exercise  18 — Oral. 

1.  Why  do  printers  require  units  of  measurement 

smaller  than  the  inch? 

2.  Into  how  many  parts  do  they  divide  an  inch  for 

the  unit  which  is  called  a  point?  How  many 
points  are  there  in  1  inch? 

3.  Into  how  many  parts  do  they  divide  an  inch  for 

the  unit  which  is  called  a  pica?  How  many 
picas  are  there  in  1  inch? 

4.  Points  in  1  inch  =  ?     Picas  in  1  inch  =  ?     How 

many  points  are  there  in  1  pica? 

5.  What  is  the  height  of  6-point  type  in  points?     In 

parts  of  an  inch? 

6.  What  is  the  height  of  8-point  type  in  points?     In 

parts  of  an  inch? 

7.  "What  is  the  height  of  12-point  type  in  points?    In 

parts  of  an  inch? 

8.  What  is  the  height  of  24-point  type  in  points?    In 

parts  of  an  inch? 

9.  What  is  the  height  of  10-point  type  in  points?    In 

parts  of  an  inch? 

10.  How  many  6-point  lines  are  there  to  an  inch  in 

the  length  of  a  column?  How  many  8-point 
Hnes?  12-point  lines?  24-point  lines?  36- 
point  lines? 

11.  How  many  picas  are  there  in  1  inch?    In  4  inches? 

In  8  inches? 

12.  Does  the  size  of  the  type  or  the  length  of  a  line 

have  anything  to  do  with  the  number  of  points 
to  an  inch?     Has  it  anything  to  do  with  the 
number  of  picas  to  an  inch? 
(VII-30) 


DENOMINATE   NUMBERS 

13.  How  long  is  a  printed  page,  if  it  is  48  picas  long? 

If  it  is  72  picas  long?  How  wide  is  it  if  it  is 
60  picas  wide? 

14.  How  many  points  high  is  a  6-point  em?     How 

many  points  wide?  How  many  6-point  ems  are 
there  in  1  inch  of  a  line  of  printing?  How  many 
6-point  ems  are  there  in  1  inch  of  a  column  of 
printing?  How  many  6-point  ems  are  there  in 
1  square  inch  of  printing? 

15.  How  many  points  high  and  wide  is  an  8-point  em? 

How  many  8-point  ems  are  there  in  1  inch  hori- 
zontally? How  many  vertically?  How  many 
are  there  in  1  square  inch? 

16.  Give  the  dimensions  of  a  12-point  em.     Tell  all 

you  can  about  it. 

17.  How  many  ems  of  6-point  type  are  there  in  a  line 

5  inches  long?  How  many  in  a  line  8  inches 
long?     12  inches  long? 

18.  How  many  ems  long  is  a  page  of  8-point  type,  if 

the  column  measures  48  picas  in  length?  60 
picas?     72  picas? 

19.  How  many  ems  of  8-point  type  are  there  in  a  line 

6  inches  long?    How  many  ems  of  24-point  type? 

20.  How  many  8-point  ems  are  there  in  a  square  inch? 

How  many  12-point  ems?  How  many  6-point 
ems? 

Exercise  19 — Oral  and  Written. 

A.  Tell  just  how  you  will  work  each  of  these  examples : 
1.  A  line  of  type  is  6|  in.  long;    how  many  points 
long  is  it?     How  many  picas  long  is  it? 
CV^I-3l) 


ARITHMETIC 

2.  A  line  of  type  is  594  points  long;  what  is  its  length 

in  inches?     In  picas? 

3.  The  Daily  News  has  a  column  22 1"  long  and  2  J" 

wide;  how  many  points  long  and  wide  is  the 
colunrn?     How  many  picas  long  and  wide? 

4.  The  type-page  of  a  magazine  is  9''  wide  and  12" 

long;  if  this  magazine  is  set  in  8-point  type, 
how  many  ems  are  there  in  one  hne?  How 
many  ems  long  is  a  column?  How  many  ems 
are  there  on  a  page? 

5.  The  type-page  of  a  book  is  36  picas  wide  and  48 

picas  long;  how  many  ems  of  12-point  type  does 
it  contain?     How  many  ems  of  6-point  type? 

6.  How  many  picas  wide  and  long  is  the  page  of  a 

book,  if  it  is  504  points  wide  and  648  points  long? 

7.  How  many  more  ems  are  there  on  a  V  X  6'^  page 

when  the  type  is  set  in  6-point  than  when  it  is 
set  in  8-point? 

8.  A  certain  page  of  12-point  type  contains  1,728  ems; 

if  the  page  is  36  picas  wide,  how  many  inches 
long  is  it? 

9.  A  certain  magazine  has  columns  2 J''  wide  and 

10 J"  long;  if  the  type-page  is  9'^  wide  and  10 J" 

long,  how  many  columns  are  there  on  a  page? 

How  many  ems  of  9-point  type  are  there  in  a 

column?     In  a  page? 
10.  What  would  be  the  length  (in  inches)  of  a  line 

set  in  8-point  type,  that  it  might  contain  as 

many  ems  as  a  line  6§"  long  set  in  6-point  type? 

How  many  picas  wide  would  it  be? 
B.  Now  work  all  the  examples. 

(VII-32) 


DENOMINATE   NUMBERS 

Lesson  10 

(For  Reference) 
Legal  Weights  of  a  Bushel 

(In  Pounds) 


United  States.  .  .  . 

Alabama 

Arizona 

Arkansas 

California 

Colorado 

Connecticut 

Delaware 

Dist.  of  Columbia 

Florida 

Georgia 

Idaho 

Illinois 

Indiana 

Iowa 

Kansas 

Kentucky 

Louisiana 

Maine 

Maryland 

Massachusetts.  .  . 

Michigan 

Minnesota 

Mississippi 

Missouri 

Montana 

Nebraska 

New  Hampshire . . 

New  Jersey 

New  York 

North  Carolina. . . 
North  Dakota.  .  . 

Ohio 

Oklahoma 

Oregon 

Pennsylvania .  .  .  . 
Rhode  Island .  .  .  . 
South  Carolina. .  . 
South  Dakota. . .  . 

Tennessee 

Texas 

\'ermont 

^'i^ginia 

Washington 

West  Virginia .  .  .  . 
Wisconsin 


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:.G 

GO 

0/ 

GO 

eo 

GO 

56 

32 

48 

,:g 

60 

60 

56 

32 

48 

56 

60 

52 

GO 

60 

60 

56 

32 

48 

56 

60 

55 

60 

CO 

GO 

56 

32 

48 

56 

CO 

52 

60 

60 

GO 

56 

32 

46 

56 

60 

CO 

56 

32 

47 

56 

56 

50 

CO 

56 

32 

48 

56 

60 

50 

60 

60 

60 

56 

32 

48 

56 

60  ' 

52 

60 

60 

60 

56 

32 

48 

56 

60 

56 

60 

60 

60 

56 

32 

48 

56 

60 

57 

60 

60 

56 

32 

48 

56 

60 

52 

62 

60 

60 

56 

30 

48 

56 

56 

57 

60 

60 

60 

56 

32 

48 

56 

60 

60 

56 

32 

48 

56 

60 

60 

60 

56 

32 

48 

56 

60 

57 

60 

60 

50 


48 


48 

45 
50 

48 

48 


44 

48 
48 
50 

48 
45 


50 

48 


50 
45 
48 


50 
45 
46 

45 

50 


(\^I-33) 


ARITHMETIC 

The  table  here  given  shows  the  number  of  pounds  of 
various  products  considered  by  law  as  constituting  a 
bushel.  It  is  not  necessary  for  you  to  learn  this  entire 
table,  but  you  should  learn  the  figures  which  appear 
opposite  your  state. 

Exercise  20 — Oral. 

In  this  state,  what  is  the  legal  weight  of: 

1.  A  bushel  of  wheat?     Of  rye?     Of  oats? 

2.  A  bushel  of  barley?    Of  shelled  corn?    Of  potatoes? 

3.  A  bushel  of  onions?     Of  beans?     Of  peas?     Of 

apples? 

4.  10  bushels  of  wheat?     Of  rye?     Of  onions? 

5.  1  pk.  of  potatoes?     Of  beans?     Of  peas? 

6.  1  qt.  of  oats? 

7.  3  pk.  of  corn? 

8.  6  bu.  of  beans? 

9.  ^  bu.  of  barley? 

10.  f  bu.  of  corn? 

11.  1  pk.  of  oats? 

12.  3  pk.  of  potatoes? 

13.  1  qt.  of  oats? 

14.  .75  bu.  of  rye? 

15.  1.5  bu.  of  barley? 

16.  .25  bu.  of  peas? 

17.  .5  bu.  of  beans? 

18.  Which  is  the  lightest  of  these  products? 

19.  Which  is  the  heaviest  of  these  products? 

20.  Pick  out  all  the  things  weighing  60  lb.  to  the  bushel. 

21.  Can  you  explain  why  there  is  so  great  a  difference 

in  weight  between  oats  and  wheat? 

(VII-34) 


DENOMINATE   NUMBERS 

Tables  of  Measures  Used  in  Certain  Professions 

(For  Information  and  Reference) 

Used  by  Surveyors  and  Civil  Engineers  for  Measuring 

Land 

Surveyors'  Linear  Measure 


7.92  inches  (in.) =  1  link  (1.) 

25       links =  1  rod  (rd.) 

4       rods =  1  chain  (ch.) 

80       chains =1  mile  (mi.) 

Surveyors'  Square  Measure 


16  square  rods  (sq.  rd.) . . .  =  1  square  chain  (sq.  ch.) 

10  square  chains =1  acre  (A.) 

640  acres =  1  square  mile  (sq.  mi.) 

1  square  mile =1  section  (sec.) 

36  sections =1  township  (T.) 

(Note  :  Point  out  the  differences  between  this  table 
and  the  Table  of  Common  Square  Measure.) 

(VII-35) 


ARITHMETIC 

Used  by  Druggists,  Chemists,  and  Physicians  for 
Measuring  Chemicals 

Apothecaries'  Liquid  Measure 


60  minims  (m.) =  1  fluid  dram  (5) 

8  fluid  drams •  =  1  fluid  ounce  (5) 

16  fluid  ounces =  1  pint  (o) 

8  pints =  1  gallon  (cong.) 

(Notice  that  1  pint  is  i  gallon  just  as  in  Common 
Liquid  Measure,  but  that  each  pint  is  divided  into 
many  smaller  parts  here,  while  in  Common  Liquid 
Measure  i  pint  or  gill  is  the   smallest  unit  used.) 


Apothecaries'  Dry  Measure 

20  grains  (gr.) =  1  scruple  (9) 

3  scruples =  1  dram  (3) 

8  drams =  1  ounce  (3) 

12  ounces =  1  pound  (lb.) 

(Note  :  Does  this  table  more  closely  resemble  Common 

Dry  Measure  or  Avoirdupois  Weight?) 

(VII-36) 


DENOMINATE   NUMBERS 


Used  by  Goldsmiths,  Silversmiths,  and  Jewelers  for 
Measuring  Precious  Metals  and  Stones 

Troy  Weight 


24  grains  (gr.) =  1  pennyweight  (pwt.) 

20  pennyweights =1  ounce  (oz.) 

12  ounces =  1  pound  (lb.) 

Exercise  21 — Oral  Review. 

1.  William  Smith  sold  merchandise  amounting  to 

$250.00  to  Frank  Jones;   who  is  the  debtor  and 
who  is  the  creditor  in  this  transaction? 

2.  Find  the  interest  on  $200.00  for  120  days  at  6%. 

At  3%.     At4i%. 

3.  Add: 

(a)        (h)  (c) 

4,634      3,872      2,798 
397       597       398 


4.  Subtract: 

(a) 

(h) 

(c) 

5,632 

2,108 

4,531 

296 

799 

195 

5.  Multiply: 

(a) 

(h) 

(c) 

24  X  16f 

72  X  66f 

(VII-37) 

48  X  87.5 

m 
ARITHMETIC 

6.  Divide: 

(a)  (h)  (c) 

15  -^  33J  800  -^  25  12  -r-  12.5 

(Practice  till  you  can  do  #3,  4,  5,  and  6  correctly  in 
from  three  to  five  minutes.) 

7.  Reduce  4  min.  20  sec.  to  minutes.     To  seconds. 

8.  How  many  inches  are  there  in  §  yard?     In  f  yd.? 

9.  How  many  square  miles  are  there  in  a  township? 

How  many  sections  are  there  in  a  township? 

10.  Say  the  table  of  Paper  Measure. 
Answer  these  rapidly: 

11.  5%of  S400.  =  ? 

12.  3%  of  $600.  =  ? 

13.  4%  of  $500.  =  ? 

14.  8%  of  $200.  =  ? 

15.  33i%  of  $300.  =  ? 

16.  12i%  of  $80.  =  ? 

17.  25%  of  $600.  =  ? 

18.  75%  of  $800.  =  ? 

Exercise  22 — Written  Review. 

1.  On  March  8,  1919,  a  boot  and  shoe  retailer  bought 
the  following  merchandise : 

48  pr.  Calf  Shoes @  $2.75 

36  pr.  Kid  Shoes @    3.00 

12  pr.  Rubbers @    1.00 

To  pay  for  this  merchandise  he  gave  a  90-day 
note  bearing  6%  interest;  what  was  the  date  of 
maturity  of  the  note?     What  amount  was  paid 
in  full  settlement  of  the  note? 
(yii-3S) 


DENOMINATE   NUMBERS 

2.  How  many  board  feet  of  lumber  are  there  in  the 

following  shipment: 

50  pieces  12'  X    8''  X  2"  =  ? 

60  pieces  10'  X  12"  X  3"  =  ? 

80  pieces    8'  X    6"  X  V  =  ? 

Total =  ? 

3.  A  salesman  received  $48.00  commission  for  selling 

a  certain  bill  of  goods;  his  commission  was 
figured  at  8%.  Make  a  good  statement  about 
$48.00  so  you  can  begin  to  work;  under  it  write 
1%  of  the  sales.     Then  100%  or  total  sales. 

4.  A  wholesale  drug  dealer  allowed  a  trade  discount 

of  30%  and  a  cash  discount  of  2%  10  days  on  all 
goods  sold  to  retail  druggists;  how  much  would 
a  retail  druggist  have  to  pay  for  a  bill  of  goods 
which  at  list  prices  amounted  to  $80.00? 

5.  What  per  cent  of  8.5  is  2.5? 

6.  A  messenger  boy  who  delivers  reports  has  40  busi- 

ness houses  on  his  route;  if  it  takes  him  3  hr. 
40  min.  to  cover  the  complete  route,  what  is  the 
average  time  required  to  deliver  each  report? 
How  many  complete  trips  can  he  make  in  44 
working  hours? 

7.  AVhat  is  the  cost  of  4M  sheets  of  28"  X  42"  book 

paper  at  $5.85  per  cwt.  on  the  basis  of  1  rm. 
42"  X  56"  =  90#? 

8.  Find  16  f%  of  10  sq.  yd.  8  sq.  ft.  6  sq.  in. 

Q.  How  many  cubic  inches  are  there  in  i  cu.  yd.? 
10.  Reduce  276,330  sec.  to  days,  hours,  minutes,  and 
seconds. 

(VII-39) 


ARITHMETIC 

Subtract,  but  do  not  copy: 
(Time  for  these  12  examples  is  less  than  4 J  minutes.) 

11.             12.             13.             14.  15.  16. 

874,362    487,398    687,481    768,042  488,398  763,904 

439,728    248,099    298,385      38,098  298,468  498,927 

17.             18.             19.             20.  21.  22. 

249,763    863,901    848,729    511,872  863,784  948,728 

152,948    248,763    385,621    355,902  497,687  349,875 


Add,  but  do  not 

copy: 

(Time  for  these  6 

examples 

is  less  than  4  J  minutes.) 

23.            24. 

25. 

26. 

27. 

28. 

4,875      48,712 

38,641 

7,741 

2,349 

1,593 

2,387           986 

864 

8,431 

7,257 

5,945 

5,963        3,639 

41,386 

9,876 

7,812 

9,198 

8,846       53,872 

64,138 

2,346 

9,136 

7,758 

5,222       91,808 

94,815 

5,966 

6,750 

9,297 

Copy  and  divide: 
(Time  for  these  6  examples  is  less  than  4^  minutes.) 

29.  10,416 -^  24;  32.     61,344  --  71; 

30.  33,867  -^  53;  33.     44,541  -^  63; 

31.  45,136  -^  62;  34.     73,882  -^  82. 

Copy  and  multiply: 
Time  for  these  8  examples  is  less  than  4 J  minutes.) 

35.  439  X  62;  39.     551  X  35 

36.  274  X  83;  40.     340  X  87 

37.  806  X  39;  41.     319  X  91 

38.  128  X  52;  42.     509  X  74. 

(VII-40) 


DENOMINATE   NUMBERS 

Lesson  11 
Special  Working  Units 


The  average  man  engaged  in  any  occupation  can,  in 
one  hour,  do  a  certain  amount  of  work;  in  two  hours 
he  can  do  twice  that  amount  of  work,  or  as  much  work 
as  two  men  can  do  in  one  hour;  thus,  it  naturally 
follows  that  when  a  piece  of  work  can  be  done,  for 
instance,  by  10  men  in  6  hours,  the  total  time  con- 
sumed by  the  10  men  in  doing  this  work  is  the  same 
as  60  hours  of  one  man's  work;  in  other  words,  the 
work  requires  60  '^ man-hours,"  a  ^^ man-hour"  being 
1  man's  effort  for  1  hour  of  time. 

The  number  of  '^man-hours"  required  for  any  piece 
of  work  is  the  product  of  the  number  of  men  engaged 
on  the  work  multiplied  by  the  number  of  hours  con- 
sumed in  doing  the  work. 

When  the  number  of  ^^ man-hours"  necessary  for  a 
piece  of  work  is  known,  we  find  either  the  num.ber  of 
men  to  be  engaged  or  the  number  of  hours  to  be  con- 
sumed, by  dividing  the  number  of  ^'man-hours"  by 

(VII-41) 


ARITHMETIC 

the  known  factor;  thus,  a  piece  of  work  which  can  be 
completed  in  24  man-hours,  can  be  done  by  1  man  in 
24  hourSj  by  2  men  in  12  hours,  by  3  men  in  8  hours, 
by  4  men  in  6  hours,  by  6  men  in  4  hours,  by  8  men  in 
3  hours,  by  12  men  in  2  hours,  or  by  24  men  in  1  hour. 

Another  ^'special  working  unit"  in  common  use  in 
railroading  is  the  ''ton-mile";  this  is  the  equivalent 
of  hauling  1  ton  a  distance  of  1  mile.  The  number  of 
tons  hauled  multiplied  by  the  number  of  miles  they 
are  hauled  equals  the  ''ton-miles." 

There  are  many  other  "special  working  units,"  such 
as  the  "kilowatt-hour,"  the  "foot-pound,"  the  "light- 
year,"  the  "acre-inch,"  etc.,  all  based  on  the  same 
general  principle  of  using  the  product  of  two  units  of 
measure. 

Exercise  23 — Oral. 

1.  When  a  certain  piece  of  work  can  be  completed  by 

2  men  in  1  hour,  how  long  would  it  take  1  man 
to  do  the  work?  How  many  man-hours  are 
needed  to  do  this  work? 

2.  How  many  man-hours  are  needed  to  do  a  certain 

job,  if  10  men  can  do  it  in  10  hours? 

3.  To  build  a  certain  wall,  5  men  worked  8  hours 

each  and  5  other  men  worked  12  hours  each; 
how  many  man-hours  were  needed  to  build  this 
wall? 

4.  To  plow  a  certain  field,  48  man-hours  of  work 

were  required;  how  many  men  should  be 
engaged  to  finish  this  work  in  8  hours?  In  6 
hours?     In  4  hours? 

(VII-42) 


DENOMINATE   NUMBERS 

5.  To  dig  a  certain  trench,  36  man-hours  of  work 

were  required;  how  quickly  can  3  men  dig  this 
trench?  4  men?  6  men?  9  men?  12  men? 
18  men?     24  men? 

6.  A  freight  engine  had  to  haul  400  tons  of  freight 

2  miles  to  place  it  on  a  side-track;  how  many 
ton-miles  were  involved  in  this  operation? 

7.  If  the  cost  of  moving  machinery  by  freight  is  lOjZ^ 

per  ton-mile,  what  is  the  cost  of  moving  10  tons 
10  miles? 

8.  The  electrical  energy  required  to  keep  1  electric 

lamp  burning  50  hours  is  equal  to  that  required 
to  keep  10  lamps  burning  for  how  many  hours? 

9.  At  1^  per  lamp  per  hour,  what  is  the  cost  of  burn- 

ing 6  lamps  5  hours? 

10.  Could  the  burning  of  1  lamp  for  1  hour  be  called 

a  lamp-hour? 

11.  Tell  in  good  English  what  is  meant  by  a  man-hour. 

12.  Tell  in  good  English  what  is  meant  by  a  ton-mile. 

Exercise  24r—  Written. 

Be  this  manufacturer  for  to-day. 

1.  You  estimated  that  it  would  require  1,500  man- 

hours  to  execute  a  contract  for  steel  rails  and 
you  had  only  10  working  days  in  which  to  do 
the  work;  if  your  men  worked  7|  hours  per 
day,  how  many  men  were  needed? 

2.  A  contractor  who  works  for  you  has  a  weekly  pay 

roll  amounting  to  $720.00;  if  his  men  work  40 
hours  per  week  and  the  rate  is  60^  per  man- 
hour,  how  many  men  has  he? 

(VII-43) 


ARITHMETIC 


3.  In  one  department  of  your  business  35  men  are 

paid  a  total  of  $2,520.  for  working  3  weeks  of 
48  hours  each;  what  is  the  rate  of  wages  per 
man-hour? 

4.  The    weekly    pay    roll    in    another    department 

amounts  to  $1,108.80;  if  the  rate  per  man-hour 
is  55^  and  there  are  48  men  employed,  how 
many  hours  per  week  do  they  work? 

5.  If  1,050  tons  of  your  freight  are  hauled  250  miles  at 

a  cost  of  1  ^^  per  ton-mile,  what  is  the  total  cost? 


6.  A  contractor  agreed  to  build  an  addition  to  your 
factory  in  16  weeks  and  estimated  that  the  work 
would  require  76,800  man-hours;  being  unable 
to  furnish  more  than  50  men  with  the  necessary 
tools  and  machinery,  he  decided  to  work  two 
shifts  of  50  men  each — a  day  shift  and  a  night 
shift.  If  the  day  shift  worked  9  hours  per  day 
6  days  a  week,  how  many  hours  per  day  for  6 
days  a  week  did  the  night  shift  work? 
(VII-44) 


DENOMINATE   NUMBERS 

7.  In  one  of  the  offices  electricity  costs  .8  cents  per 

lamp  per  hour;  how  many  lamps  can  be  burned 
30  hours  for  $9.60? 

8.  How  long  could  50  lamps  be  burned  for  $22.50, 

if  the  cost  per  lamp  per  hour  were  .75  cents? 


(Yn-45) 


FRACTIONS 


Lesson  12 
Fractions  as  You  Want  to  Know  Them  Now 
Exercise  25 — Oral. 
Answer  at  sight: 

1.  i  of  i  = 

2.  t  of  f  = 

3.  i  of  I  = 

4.  i  of  I  = 


5.  I  of 


13. 
14. 
15. 
16. 


2  plus 
1  + 


of  i 


6. 

7. 
8. 

:     ? 


of 


*of 


8  __  9 

9  —     i 
=    ? 


15 


8.    _     ? 
9    —     . 


I  of  -^  =  ? 


9. 
10. 
11. 
12. 


1    Qf    49 


f  of 

7 
10 


of 
of 


50     — 

4.    

5     — 


30 
35 

27 
5 


? 
=  ? 
=  ? 
=     ? 


iof|  = 


17.  4  of  i  added  to  f  of  12  =  ? 

18.  f  of  I  less  i  of  i  =  ? 


1  minus  i 


of 


of  i  =  ?      19.  i  of  U  plus  i  of  2i  =  ? 
i)  =  ?  20.  f  of  i  +  5i  +  6i  =  ? 

21.  What  did  you  add  to  2  in  example  #13?     2  +  i 

of  i  =  ?    Read  it ;  it  will  tell  you  just  what  to  do. 

22.  When  two  or  more  steps  occur,  what  is  done  first? 

Example :  i  of  i  added  to  |  of  12  =  ? 


Exercise  26 — Written. 


EXAMPLE:     If  6  yd.  of  goods  cost  $1.50,  what  will  one  yd.  cost' 
$li.  -^  6  =  i  of  $li.,  or  $1.,  Ans. 


1. 

4i- 

-    3  or  ?  of  ?  =  ? 

6. 

lOi  - 

-    7  =  ? 

2. 

H- 

-    7  or  ?  of  ?  =  ? 

6t- 

-    5  =  ? 

3. 

2i- 

-    2  or  ?  of  ?  =  ? 

8. 

7i- 

-  12  =  ? 

4. 

n- 

-  11  or  ?of  ?  -  ? 

9. 

9i- 

-  14  =  ? 

5. 

20i  - 

-    7  or  ?  of  ?  =  ? 

10. 

101  - 

-    9  =  ? 

\ 


FRACTIONS 


Another  way  to 

see  the 

same  step : 

EXAMPLE: 

Dividend  1^ 
Divisor      6 

=  U  -^6 

or 

1 

4j 

Ans. 

EXAMPLE 

^ 

•      5    -^ 

-  5, 

or  1  X  h 

or 

'.t 

Ans. 

1 JL  Qi. 

11.       ii  =  ?  12.     ^  =  ? 

2  7 

13.     lOi.,  14.     i^  =  ? 

3  2 

15.     -i  =  ?  16.     -  *  =  ? 

4  12 


EXAMPLE:     If  1^  yd.  cost  $6.00  what  will  one  yd.  cost 

Dividend  S6.      «^        .,,         a,o    ^,  -i        o-a     \ 
T^.   .  -T-=  $6.  -^  1|,  or  $6.  X  f,  or  $4.,  Ans. 

Divisor      1  h 


17.   1  =  ?     18.  ^  =  ? 

2i  3i 

19.  —  =  ?     20.  4  =  ? 

lOi  3i 

21.  60  +  ^  =  ?  22.  600  +  —  =  ? 
5  50 


880  110   ^ 

23   —  =  '^             24  —  =  ^ 

^'^'            22      •          ^^'  2_2 

(VII-47) 


ARITHMETIC 


EXAMPLE:     ^  =  ? 
6 

If  _  1.75       175        7 

6    ~  6.00 '''■  600 '''"  24' ^''^• 


Very  often  fractions  of  this  kind  can  be  cleared  most 
easily  by  changing  both  terms  to  decimals  of  a  like 
number  of  places  and  reducing  to  lowest  terms. 

25.     —  =  ?  26.     -  =  ? 

38 

27.     —  =  ?  28.     —  =  ? 


74 

? 

2i_ 
50 

? 

H  . 

=  ? 

26i 

in  '"    s 


29.     --  =  ?  30.     —  =  ? 


EXAMPLE: 

1^  _  1^  X  2        3        J    .        (We  use  2  as  the  number  to  mul- 
~6    ~  '6"^2"  ^^  12  ^^  ^'  ^^^-       tiply  by,  because  2  is  the  de- 
nominator of  the  fraction  |.) 


Since  the  quotient  in  an  example  in  division  is  not 
changed  by  multiplying  the  dividend  and  the  divisor 
by  the  same  number,  we  can  multiply  the  numerator 
and  the  denominator  of  a  fraction  by  the  same  number 
without  changing  the  value  of  the  fraction. 


EXAMPLE: 

1  f  _  1  f  X  6       10       2     .         (We  use  6  as  the  number  to  mul- 
7|~7^X6      45       9'        '       tiply  by,  because  6  is  the  least 

common    denominator    of    f 

and  5.) 


(VII^8) 


FRACTIONS 

When  either  term  of  a  fraction  contains  a  fraction,  as 

1-  4- 

7?'  qT'  ^^^-j  ^^^  whole  is  called  a  "complex  fraction. '^ 

When  both  terms  of  a  complex  fraction  contain  frac- 
tions, clear  by  multiplying  both  terms  by  that  number 
which  would  be  the  least  common  denominator  of  the 
two  fractions. 

31.     ^  =  ?  32.     ^  =  ? 

4  5i 

33.     ^  =  ?  34.     ^  =  ? 

4i  6i 

1-  2- 

35.     —  =  ?  36.     —  =  ? 


4i  1 


37.     —  =  ?  38.     —  -  ? 


A3  O3 

^  =  ?  40.     ^ 

5i  2i 


2-  *^- 

39.     —  =  ?  40.     —  =  ? 


(VII-49) 


MULTIPLICATION 

Lesson  13 
Cross  Multiplication 

"Cross  Multiplication"  is  a  method  of  multiplica- 
tion in  which  it  is  unnecessary  to  write  any  partial 
products,  as  the  final  product  is  obtained  at  once. 
In  multiplying  numbers  of  two  orders  as  23  X  21,  it 
23     will  be  noticed  that  the  units'  figure  of  the  product 
21     is  obtained  by  multiplying  the  units'  figures  of  the 
23     multiplier  and  of  the  multiplicand ;  thus,  3X1=3. 
46       The  tens'  figure  of  the  product  is  the  sum  of  the 
483     tens'  figure  of  the  multiplicand  multiplied  by  the 
units'  figure  of  the  multiplier,  plus  the  units'  figure  of 
the  multiplicand  multiplied  by  the  tens'  figure  of  the 
multipUer;  thus,  (2  X  1)  +(3  X  2)  =  8.  The  hundreds' 
figure  of  the  product  is  obtained  by  multiplying  the 
tens'  figures  of  the  multiplicand  and  of  the  multiplier; 
thus,  2X2  =  4. 

Now,  if  we  had  added  2  and  6  in  tens'  place  without 
^>2Z^  writing  them,  we  would  have  had  483  for  the 
S^4 —   answer  just   as  before,   but   we  would   have 
483       written  no  partial  products;  thus: 
3X1=3;  write  3; 

2  X  1  =  2;  3  X  2  =  6;  2  +  6  =  8;  write  8; 
2X2  =  4;  write  4. 
Hemember,  we  multiply  in  this  order: 
Fu-st:  Units  X  Units.  23^ 

21-^ 
3 

(VII-50) 


]MULTIPLICATION 

Second:  (Units  X  Tens)  plus  (Tens  X  Units).   \.^ 

Third:  Tens  X  Tens. 


EXAMPLE:     34  X  26  =  884. 
34  First:  4  X  6  =  24;  write  4  and  carry  2; 

26  Second:    3X6  =  18;  4X2  =  8;  18+8  +  2  (car- 

884  ried)  =  28;  write  8  and  carry  2; 

Third:  3X2=6;  6+2  (carried)  =  8;  write  8. 

EXAMPLE:     47  X  35  =  1,645. 
47         First:  7  X  5  =  35;  write  5  and  carry  3; 
35        Second:  4  X  5  =  20;   7  X  3  =  21;  20  +  21  +  3 
1,645  (carried)  =  44;  write  4  and  carry  4; 

Third:  4  X  3  =  12;  12  +  4  (carried)  =  16;  write  16. 


If  there  is  a  carrying  figure  from  any  product  or  sum, 
carry  to  the  next  place  as  usual. 

Exercise  27 — Oral. 

1.  In  the  multiplication  here  shown,  how  is  the       14 

units'  figure  of  the  product  obtained?  21 

2.  Show  how  the  tens'  figure  of  the  product  is       14 

obtained?     Point  it  out.  28 

3.  Tell  us  where  the  hundreds'  figure  of  the    294 

product  comes  from. 

4.  Begin  the  multiplication.     No  writing  of  partial 

products. 

43 
X21 

(VII-51) 


ARITHMETIC 

5.  Show  where  you  get  the  tens'  figure  of  the  product. 

6.  Show  where  you  get  the  hundreds'  figure  of  the 

product. 

7.  What  is  done  with  the  carrying  figures  in  cross 

multipHcation? 

Exercise  28 — Written. 

Multiply    the    following    without    writing    partial 

products : 

1.     31  2.     23  3.     32  4.     42 

22  21  21  21 


5. 

24 

6. 

32 

7. 

42 

8. 

42 

22 

24 

24 

34 

9. 

42 

10. 

43 

11. 

46 

12. 

32 

26 

34 

33 

27 

Do  not  copy  the  following  to  multiply;  write  nothing 
but  the  answers: 

13.  43  X  32;  16.  98  X  31 

14.  28  X  31;  17.  89  X  22 

15.  37  X  23;  18.  48  X  41 

19.  76  X  34; 

20.  84  X  42. 

21.  Find  the  cost  of  the  following  merchandise,  with- 
out writing  partial  products : 

45#  Rice @     14^  per  lb.  .  .  .$?.?? 

24  Cans  Peaches  .  @     17j^  per  can . 
14#  Cocoa @     3S^per  lb.  . 


Total I?.?? 

(VII-52) 


MULTIPLICATION 


22.  72  X  18;        25.  82  X  15 

23.  21  X  22;         26.  79  X  21 

24.  65  X  32;        27.  93  X  14 

(Practice  this  till  you  can  do  the  last  nine  in  5  minutes 

or  less.) 


28.  93  X  31; 

29.  97  X  15; 

30.  64  X  64. 


VIi-53>j 


TIME   AND   WAGES 

Lesson  14 

How  Wages  Are  Figured 

Most  tradesmen  are  paid  by  the  liour  (or  fraction  of 
an  hour)  for  their  labor,  based  on  a  certain  sum,  as 
S25.00,  $30.00,  etc.,  for  a  week's  work  consisting  of  a 
certain  number  of  hours,  as  42,  45,  48,  etc.;  thus,  if  a 
man  is  working  on  a  48-hr.  basis  at  $25.00  per  week,  he 
receives  ts  of  $25.00,  or  52r2f!^  for  every  hour  he  works, 
and  his  wages  for  46  hr.  would  be  ff  of  $25.00,  or  $23.96. 


EXAMPLE: 

12  hr.   @  $24.00  per  week  on  a  48-hr.  basis  =  ? 
if  =  i;  i  of  $24.00  =  $6.00,  Ans. 

Always  use  aliquot  parts  when  possible,  as: 
8  hr.  =  A  or  i  of  a  48-hour  week; 
12  hr.  =  if  or  T  of  a  48-hour  week;  etc. 


EXAMPLE:     9  hr.  @  $30.00  per  week  on  a  48-hr.  basis  - 

8  hr.  =  I  wk.;  i  of  $30.00  =  $5.00; 
l_hr.  =  I  of  8  hr.;  |  of  $5.  =    0.63; 

9  hr.  =  $5.63,  Ans. 


When  the  number  of  hours  worked  is  an  aliquot  part 
of  a  week  plus  or  minus  a  fraction  of  such  ahquot 
part,  find  the  amount  corresponding  to  the  aliquot  part 
and  add  or  subtract  the  amount  corresponding  to  the 
fraction  of  the  aliquot  part. 

(VTI-54i 


TIME   AND   WAGES 


EXAMPLE:     43  hr.  @  $25.00  per  week  on  a  48-hr.  basis  =  ? 
If  of  $25.00  =  %^£p; 

$22.40 
48)  $1075.00     Ans.,  $22.40 


When  the  use  of  aliquot  parts  is  impossible,  find  the 
proper  fractional  part  of  the  rate  per  week,  but  always 
multiply  by  the  numerator  before  you  divide  by  the 
denominator  because  you  will  usually  have  to  multiply 
a  difficult  fraction  if  you  first  divide  by  the  denominator 
to  find  the  rate  per  hour  and  then  multiply  by  the 
numerator. 


EXAMPLE:     42f  hr.  @  $27.00  per  week  on  a  48-hr.  basis  =  ? 

9 
421  _  42.75.  4275      2700  _  38475 

"48  -  48:00'  im^~r-    le  ^'  ^^^•^^'  '^'''• 

16 

57 
42!  _  42f  X  4   171.  X7X      2700  _  153900 
"48  -  ISX^''"  19-2'  m^^r  -      64   ^'  ^^^•^^'  ^"'• 

64 


421     ^ 
When  fractions  of  an  hour  are  given  as  -— -,  clear  by 

4o 

multiplying  both  terms  of  the  fraction  by  4  and  con- 
tinue as  before,  or  clear  by  using  decimals. 

Examine  this  and  state  which  is  the  shortest  way. 

Exercise  29 — Oral. 

1.  If  a  carpenter  is  paid  on  the  basis  of  45  hr.  per 
week,  what  fraction  of  a  week's  wages  does  he 
receive  for  1  hour's  work? 

(VII-55) 


ARITHMETIC 

2.  A  machinist  is  paid  on  the  basis  of  48  hr.  per  week 

what  fraction  of  a  week's  wages  does  he  receive 
for  23  hours'  work? 

3.  An  electrician  is  paid  on  the  basis  of  42  hours  per 

week;  what  fraction  of  a  week's  wages  does  he 
receive  for  39  J  hours'  work? 

4.  At  the  rate  of  $32.00  for  48  hours'  work,  how 

would  you  find  the  amount  of  wages  to  be  paid 
to  a  bricklayer  for  working  8  hours? 

5.  A  teamster  who  is  paid  on  the  basis  of  $24.00  for 

52  hours'  work  per  week,  worked  47 J  hours; 
how  would  you  find  the  amount  due  him? 

6.  When  aliquot  parts  cannot  be  used,  why  is  it 

easier  to  multiply  by  the  numerator  first  and 
then  divide  by  the  denominator,  than  it  is  to 
divide  first  to  find  the  rate  per  hour  and  then 
multiply  to  find  the  amount  for  a  certain  number 
of  hours? 

7.  What  aliquot  part  of  48  hours  is  6  hours?    8  hours? 

12  hours?     16  hours?    24  hours? 

8.  What  aliquot  part  of  42  hours  is  6  hours?    7  hours? 

14  hours?     21  hours? 

9.  What  aliquot  part  of  45  hours  is  9  hours?     15 

hours?  18  hours?  27  hours?  30  hours?  36 
hours? 

10.  When  the  number  of  hours  worked  is  an  aliquot 

part  of  the  number  of  hours  per  week,  what  is 
the  easiest  way  of  finding  the  amount  of  wages 
to  be  paid? 

11.  How  can  you  find  the  wages  for  9  hours  on  the 

basis  of  48  hours  per  week,  using  aliquot  parts? 
How  for  7  hours? 

(VII-56) 


TIME   AND   WAGES 

12.  How  can  you  find  the  wages  for  10  hours  on  the 
basis  of  48  hours  per  week?     For  13  hours? 

Exercise  30 — Written. 
Mr.  Rice's  Problems  Concerning  His  Men's  Wages. 


1.  In  Department  "A''  the  men  are  paid  on  the  basis 

of  48  hr.  per  week;   how  much  money  would  be 
paid  to  each  of  the  following  men: 

Adams  worked  45  hr.  @  $18.00  per  week. 
Jones    worked  48  hr.  @  $24.00  per  week. 

2.  In  Department  *^B''  the  men  are  paid  on  a  48-hr. 

basis;  how  much  money  did  he  pay  to  each  of 
the  following  workers: 

Bailey  worked    8  hr.  @  $21.00  per  week. 
Davis  worked  10  hr.  @  $24.00  per  week. 

3.  In  Department  "C"  where  the  men  are  paid  on  a 

45-hr.  basis,  what  would  be  the  total  amount  of 
the  following  pay  roll : 

(VII-57) 


ARITHMETIC 

2  men  worked  45  hr.  each  @  $30.00  per  week. 

4  men  worked  45  hr.  each  @  SI 8.00  per  week. 

3  men  worked  42  hr.  each  @  $15.00  per  week. 
6  men  worked  30  hr.  each  @  $21.00  per  week. 

5  men  worked  30  hr.  each  @  $24.00  per  week. 

4.  S.  Smith  works  in  one  of  the  departments  where 

the  working  hours  are  from  8.00  to  12.00  o'clock 
A.  M.  and  from  12.30  to  5.00  o'clock  p.  m.  from 
Monday  to  Friday  inclusive,  and  from  8.00 
A.  M.  to  1.30  p.  M.  on  Saturday  (without  a  stop 
for  lunch) ;  how  much  would  Smith  receive  for 
working  36  hr.  @  $24.00  per  week? 

5.  J.  Brown  went  to  work  at  $8.00  per  week  in  the 

shop  where  the  working  hours  are  from  8.30  to 
12.00  A.  M.  and  from  1.00  to  5.00  p.  m.  from  Mon- 
day to  Friday  inclusive,  and  from  8.30  A.  M.  to 
1.00  p.  M.  on  Saturday;  how  much  would  he 
receive  for  working  full  time  from  Monday 
morning  to  Friday  night? 

6.  At  the  rate  of  $24.00  for  48  hours'  work,  how 

many  hours  must  James  Fitzgerald  work  to 
receive  $20.00? 

7.  At  the  rate  of  $33.00  for  45  hours'  work,  how 

many  hours  must  Samuel  Jones  work  to  receive 
$27.50? 

8.  If  Tom  Daly  is  paid  $15.00  for  working  20  hours, 

how  much  would  he  receive  for  working  a  full 
week  consisting  of  48  hours? 

9.  At  what  rate  of  wages  for  a  week  of  48  hours  must 

Wm.  Beach  work  to  receive  $7.50  for  18  hours' 
work? 

(VII-58) 


TIME   AND   WAGES 

Lesson  15 
Transposition  in  Figuring  Wages 


EXAMPLE: 

7\  hr.  @  $12.00  per  week  on  a  48-hr.  basis.  =  ? 

7|  hr.  @  $12.00  =  12  hr.  @  $7.25;, 

H  =  i;  i  of  $7.25  =  $1.81,  Ans. 

We  can  ' 'transpose"  or  change  the  order  of  the 
numerators  in  an  example  in  cancellation  without  affect- 
ing the  result,  as  is  very  readily  proven.  By  using 
this  simple  device  in  figuring  wages,  much  time  and 
work  can  be  saved. 

Is  2  X  4  the  same  as  4  X  2?  Is  10  X  12 1  the  same 
as  12h  X  10? 

Supposing  we  wanted  to  find  the  amount  of  wages  to 
be  paid  for  37|  hours'  work  at  $16.00  per  week  on  a 
basis  of  48  hours  per  week;  if  we  transpose  the  hours 
worked  and  the  rate  per  week,  the  example  would  read : 
16  hours  @  $37.25  per  week  on  a  48-hour  basis,  and 
could  be  worked  very  easily  by  aliquot  parts  as  follows : 

if  =  i  week;    i  of  $37.25  =  $12.42,  which  is  the 

answer.     We  chose  H  because  16  is  an  aliquot  part  of 

48.     Watch  for  the  relation  of  one  number  to  another. 

16  37^ 

—  of  $37.25  gives  the  same  answer  as  — -  of  $16.00. 

48  48  *     * 

Therefore,  when  the  rate  per  week  is  an  aUquot  part 
of  the  weekly  hour  basis,  call  the  hours  ^'dollars''  and 
the  dollars  '' hours''  and  work  as  usual.  In  this  way, 
wages  for  almost  any  number  of  hours  at  $6.00,  $8.00, 
$12.00,  etc.,  per  week  on  a  48-hour  basis  can  be  figured 
by  the  use  of  aliquot  parts  without  using  paper. 

(VII-59) 


ARITHMETIC 

Exercise  31 — Oral. 

1.  What  is  meant  by  transposition? 

2.  Transpose  these  numbers:  8,  12. 

3.  When  the  rate  per  week  is  $8.00  and  the  basis  is 

48  hours,  how  can  we  simphfy  the  work  of  find- 
ing the  wages  for  37|  hr.? 

4.  When  the  basis  is  45  hr.  per  week,  and  the  rate  is 

$15.00,  how  can  we  find  the  wages  for  41 J  hours 
quickly? 

5.  What  amount  of  wages  should  be  paid  for  30  hours' 

work  at  $22.00  per  week  on  a  44-hour  basis? 

Exercise  32 — Oral  and  Written. 

A.  Tell  how  to  work  each  of  the  following  examples: 

On  the  basis  of  48  hours  per  week,  what  amount  of 
wages  should  be  paid  for: 

1.  16  hr.  @  $20.00  per  week? 

2.  20  hr.  @  $16.00  per  week? 

3.  37 1  hr.  @  $24.00  per  week? 

4.  40|  hr.  @  $12.00  per  week? 

On  the  basis  of  45  hours  per  week,  what  amount  of 
wages  should  be  paid  for: 

5.  15  hr.  @  $22.00  per  week? 

6.  22  hr.  @  $15.00  per  week? 

7.  26i  hr.  @  $22.50  per  week? 

8.  37.5  hr.  @  $30.00  per  week? 

On  the  basis  of  42  hours  per  week,  what  amount  of 
wages  should  be  paid  for: 

9.  36f  hr.  @  $21.00  per  week? 

(VII-60) 


TIME    AND   WAGES 

10.  4|  hr.  @  $14.00  per  week? 

B.  Now  work  all  of  them,  but  do  as  much  of  the 
work  as  possible  mentally. 

Exercise  33 — Oral  Review. 

1.  Find  the  interest  on  $300.00  for  30  days  at  6%. 

For  60  days  at  5%. 

2.  Add: 

(a)  (h)  (c) 

3,687  1,325  1,498 
498                 897  867 


3. 

Subtract : 

(a) 

(h) 

(c) 

3,675 

4,722 

3,621 

398 

699 

295 

4. 

Multiply: 

(a) 

(h) 

(c) 

72  X  16| 

32  X  25 

96  X  12.5 

5. 

Divide : 

(a) 

(h) 

(c) 

12  --  33i 

600  -  25 

400  -  16| 

(Time  yourself  on  #2,  3,  4,  and  5  and  state  time  used.) 

6.  How  many  square  miles  are  there  in  a  township? 

How  many  sections? 

7.  A  certain  wall  was  built  by  8  men  in  12  hours; 

how  many  man-hours  of  work  were  required? 
How  many  men  would  have  been  able  to  build 
this  wall  in  6  hours? 

(VII-61) 


ARITHMETIC 

8.  Multiply  without  writing  partial  products : 

(a)  (6)  (c) 

67  25  89 

23  15  31 

9.  At  the  rate  of  $28.00  for  48  hours'  work,  how 

much  should  be  paid  to  a  conductor  who  worked 
36  hours? 

10.  At  the  rate  of  $36.00  for  48  hours'  work,  how  much 

should  be  paid  to  a  motorman  who  worked  28 
hours? 

11.  What  is  the  interest  on  $75.  for  60  days  at  6%? 

Exercise  34 — Written  Review. 

1.  A  furniture  dealer  bought  the  following  merchan- 

dise from  a  manufacturer  on  April  16,  1918: 

24  chau-s  @  $5.25 

12  tables   @  18.00 
6  lamps  @    8.33i 
To  pay  for  this  merchandise,  he  gave  a  note 
bearing  5%  interest,  due  in  120  days.     What 
was  the  date  of  maturity  of  the  note?     What 
amount  was  paid  in  full  settlement? 

2.  A  real  estate  broker  received  a  commission  of 

$600.00  for  selling  a  piece  of  property;  if  his 
commission  was  figured  at  6%,  what  was  the 
amount  of  sale? 

3.  What  per  cent  of  48.75  is  32.5? 

4.  What  is  the  cost  of  3,500  sheets  of  21''  X  28" 

book  paper  @  $6.00  per  cwt.  on  the  basis  of 
42"  X  56"  =  92#  per  ream? 
(VII-62) 


TIME   AND   WAGES 

5.  How  many  board  feet  of  lumber  are  there  in    the 

following  shipment : 

60  pieces    8'  X    6"  X  2" 

75  pieces  12'  X  10''  X  1" 
100  pieces  12'  X  4"  X  2" 
How  much  is  this  lumber  worth  @  $35.00  per  M.? 

6.  I  is  what  %  of  to? 

7.  The  men  employed  by  a  certain  contractor    work 

on  this  schedule:  5  days  per  week,  8.00  to  12.00 
A.  M.;  12.30  to  5.15  p.  m.;  Saturday,  8.00  a.  m. 
to  12.15  p.  M.;  how  many  hours  per  week  must 
these  men  work  to  receive  full  time?  How  much 
would  a  man  working  for  $24.00  per  week  lose 
by  being  absent  from  work  one  afternoon? 

Add,  but  do  not  copy: 

(Time  for  these  7  examples  is  less  than  4|  minutes.) 


8. 

9. 

10. 

29,841 

248,940    486,312 

476 

37,297 

449 

97 

8,767 

86,498 

480 

542 

73,812 

9,520 

71 

419 

497 

487 

98 

48 

97,282 

586 

776 

741 

138 

27,987 

72 

45 

11. 

12.       13. 

14. 

48,982 

92,872    92,349 

97,431 

27,911 

28,947    21,572 

87,287 

88,498 

47,926    88,472 

27,478 

(VII-63) 


ARITHMETIC 


Subtract,  but  do  not  copy: 
(Time  for  these  12  examples  is  less  than  4|  minutes.) 

15.  16.  17.  18. 

768,175    943,275    429,381    676,275 
648,987    796,387    329,489    487,386 


19. 

978,947 

784,958 

23. 

289,177 
192,812 


20. 

272,983 
193,884 

24. 

892,947 
496,363 


21. 

913,462 
483,493 

25. 
285,102 
97,570 


22. 

487,896 
279,997 

26. 

484,801 
379,989 


Copy  and  multiply: 
(Time  for  these  6  examples  is  less  than  4§  minutes.) 


27.  4,746  X  97 

28.  9,728  X  49 

29.  2,864  X  86 


30.  9,796  X  63; 

31.  8,632  X  87; 

32.  124  X  36. 


Copy  and  divide: 
(Time  for  these  15  examples  is  less  than  4|  minutes.) 


33.  1,269  -^  47 

34.  2,736  -i-  38 

35.  864  -T-  24 

36.  1,316  -^  47 

37.  918  -^  34 


38.  4,465  -^  95 

39.  1,564  -T-  46 

40.  729  ^  27 

41.  644  -r-  23 

42.  884  -^  26 


43.  1,222  -T-  47 

44.  9,409  -^  97 

45.  918  4-  27 

46.  864V  36 

47.  1,620  -^  45 


(VII-64) 


MENSURATION 

Lesson  16 
The  Circle 


A  circle. 


A  circle  divided 
into  two  semi- 
circles by  a  di- 
ameter. 


A  circle  divided 
into  four  quad- 
rants by  two 
diameters  drawTi 
at  right  angles. 


A  circle  showing 
two  radii,  a  sec- 
tor, and  an  arc. 


A  ^^ circle"  is  a  plane  figure  bounded  by  one  continu- 
ous curved  line,  which  at  all  points  is  a  uniform  distance 
from  a  point  in  the  center  of  the  figure. 

The  distance  around  a  circle  is  its  '^ circumference" 
just  as  the  distance  around  a  square  or  oblong  is  its 
'^perimeter." 

A  straight  line  drawn  from  one  point  in  the  circum- 
ference to  another  point  in  the  circumference  through 
the  center  is  a  ^^ diameter."  A  diameter  divides  a  circle 
into  two  halves,  each  of  which  is  called  a  "semi-circle," 
"semi"  meaning  hah. 

Two  diameters  drawn  at  right  angles  to  each  other 
divide  a  circle  into  four  quarters,  each  quarter  being 
called  a  "quadrant." 

A  straight  line  drawn  from  the  center  to  a  point  in 
the  circumference  is  called  a  "radius,"  and  several  such 
Hues  are  called  "radii." 

(\7I-65) 


ARITHMETIC 

Any  part  of  the  circumference  of  a  circle  is  called  an 
''arc/'  and  the  figure  bounded  by  two  radii  and  an  arc 
is  called  a  '' sector '^  of  the  circle. 

Exercise  35 — Oral. 

Go  to  the  board  to  show  all  of  these: 

1.  Be  ready  to  draw  a  circle. 

2.  Draw  the  diameter  or  ''through  line"  of  this  circle 

and  say  how  long  it  is. 

3.  Into  how  many  parts  is  the  circle  now  divided? 

What  is  each  of  these  parts  called? 

4.  Draw  another  diameter  at  right  angles  to  the  first 

diameter.  How  long  is  this  second  diameter? 
Into  how  many  parts  is  the  circle  now  divided? 
Name  each  part. 

5.  Rule  two  radii  so  that  the  ruler  will  show  a  distance 

of  1  inch  between  the  points  where  the  two  radii 
touch  the  circumference.  What  is  the  length  of 
each  of  these  radii? 

6.  What  is  the  part  of  the  circumference  between  the 

two  radii  called?  What  is  the  remainder  of  the 
circumference  called? 

7.  What  is  the  figure  bounded  by  the  two  radii  and 

that  part  of  the  circumference  which  lies  between 
the  two  radii,  called? 

8.  What  is  a  circle?     What  is  half  a  circle  called? 

What  is  a  quarter  of  a  circle  called? 

9.  What  is  the  curved  boundary  of  a  circle  called? 
10.  What  is  a  line  dra^vn  through  the  center,  touching 

the  circimiference  at  two  opposite  points,  called? 
Such  a  line  divides  the  circle  into  what? 
(VII-66) 


MENSURATION 

11.  What  is  a  line  drawn  from  the  center  to  the  cir- 

cumference of  a  circle  called?     Give  one  word 
for  many  of  them. 

12.  WTiat  is  any  part  of  the  circumference  of  a  circle 

called? 

13.  What  is  the  figure  bounded  by  two  radii  and  an 

arc  called? 

14.  How  many  sides  has  it? 

Lesson  17 
The  Ratio  of  the  Circumference  to  the  Diameter 

Measure  the  circumference  and  diameter  of  a  coin, 
a  wheel,  a  plate,  and  a  spool.  Arrange  them  in  order. 
Compare  each  C  with  its  D. 

(Coin)  C:D=? 
(Wheel)  C  :  D=  ? 
(Plate)  C  :  D=  ? 
(Spool)   C  :  D=  ? 

Write  the  ratio  for  each  of  these.  Is  the  ratio  about 
the  same  each  time?  What  do  we  know  is  the  relation 
of  C  to  D? 

The  ratio  shown  by  your  comparison  is  a  little  over  3, 
or  nearly  3.1416;  therefore,  for  all  ordinary  purposes, 
3.1416  or  3t  will  be  used.  This  ratio  is  expressed  by 
the  Greek  letter  tt  which  is  called  ^^pi"  (pronounced  pi). 

By  using  the  letter  C  to  represent  the  circumference, 
D  for  diameter,  R  for  radius,  and  x  for  pi  or  3.1416, 
we  can  show  that  the  circumference  equals  the  diameter 
multiplied  by  pi  in  this  manner: 

C    =    D    X     TT. 

(VII-67) 


ARITHMETIC 

Making  a  statement  in  this  form  is  called  an  ^*  equa- 
tion." An  equation  is  a  statement  showing  the  equality 
of  two  quantities  by  placing  one  before  and  one  after 
an  equality  sign. 

Exercise  36 — Oral. 

1.  Express  the  ratio  3.1416  as  a  common  fraction 

(approximately) . 

2.  If  you  have  D,  what  must  you  do  to  get  C? 

3.  If  you  have  C,  how  would  you  find  D? 

4.  If  the  diameter  of  the  driving  wheel  on  a  locomo- 

tive is  10  feet,  tell  how  to  find  the  circumference 
of  the  wheel.  How  many  feet  would  this  wheel 
travel  on  the  track  in  making  one  revolution? 

5.  Express  the  relation  of  D  to  R. 

6.  How  can  we  find  the  radius  when  the  diameter  is 

known? 

7.  How  can  we  find  the  radius  when  the  circumfer- 

ence is  known? 

8.  How  can  we  find  the  circumference  when  the  radius 

is  known? 

9.  If  C  stands  for  circumference,  D  for  diameter,  R 

for  radius,  and  t  for  pi,  read  the  following 
equations: 

(c)  D  =  2R;  p  _  p. 

(d)  C  =  2R  X  tt;  ^^^  ^       2' 

This  is  a  very  short  way  of  telling  your  rules. 
10.  Using  equations,  make  statements  to  show: 
(a)  That  the  diameter  equals  twice  the  radius; 

(VII-68) 


MENSURATION 

(h)   That  the  diameter  equals  the  circumference 

divided  by  pi; 
(c)    That  the  circumference  equals  the  diameter 

multiplied  by  pi; 
(c?)  That  the  circumference  equals  twice  the  radius 

times  pi; 
(e)   That  the  radius  equals  one-half  the  diameter; 
(/)  That  the  radius  equals  one-half  of  the  quotient 

of  the  circumference  divided  by  pi; 

Exercise  37 — (a)  Oral.     Tell  how  to  work  each  of  the 

following  examples: 
(6)  Written.      Find    the    circumference, 
diameter,  or  radius  as  required : 

4.  C  =  20.4204  in. ;  D  =  ? 

5.  R  =  10  ft.;  C  =  ? 

6.  C  =  25.1328  in.;  R  =  ? 

7.  Find  the  circumference  of  a  table  which  has  a 

diameter  of  1  yd. 

8.  Find  the  radius  of  a  circular  lake  which  has  a 

diameter  of  100  yd. 

9.  Find  the  diameter  of  a  circle  which  has  a  circum- 

ference of  9  yd.     Carry  it  to  2  decimal  places. 
10,  Find  the  circumference  of  a  circle  which  has  a 
radius  of  1  yd.  2  ft.  3  in. 

Exercise  38 — Written. 

1.  Find  the  circumference  of  Tom's  bicycle  wheel  if 

the  diameter  is  28".     (Use  3t  for  pi.) 

2.  The  spokes  in  a  certain  wheel  are  2  in.  apart  at 

the  rim;    what  is  the  diameter  of  the  wheel  if 
there  are  45  spokes?     (Use  3.1416) 
(VII-69) 


1.  D 

=  4  in.;    R  = 

2.  R 

=  3  ft.;    D  = 

3.  D 

=  4  yd.;  C  = 

ARITHMETIC 


3.  In  the  same  wheel,  what  is  the  length  of  each  spoke 

from  the  rim  to  the  hub  of  the  wheel,  if  the  hub 
is  2.64  in.  in  diameter? 

4.  The  circumference  of  a  locomotive  wheel  is  24  ft. ; 

how  often  must  it  revolve  while  the  locomotive 
travels  1  mile? 

5.  The  cogs  on  a  cog  wheel  are  |  in.  apart;    what 

radius  must  the  wheel  have  if  there  are  224  cogs? 
(Use  3.1416) 

6.  The  drum  of  a  windlass  has  a  diameter  of  2  ft.; 

how  many   times  must   the   drum  revolve   to 
move  a  block  of  granite  44  ft.?     (Use  3+.) 

7.  A  circular  running-track  has  a  diameter  of  84.033 

ft.;   how  many  laps  must  one  run  on  this  track 
to  cover  a  mile?     (Use  3.1416) 

8.  In  running  a  100-yard  dash  on  this  track,  how 

many  complete  laps  and  how  many  yards  in 
addition  would  one  have  to  run? 

Lesson  18 
Finding  the  Area  of  a  Circle 


^qOHFE^^ 


CIRCUMFERENCE    OF  CIRCLE 

(VII-70) 


MENSURATION 


If  you  divide  a  circle  into  sectors  and  cut  them  apart 
as  shown  in  the  illustration,  you  will  find  that  each  sec- 
tor would  be  a  triangle,  were  it  not  for  the  fact  that 
it  has  a  slightly  curved  base  instead  of  a  straight  base. 

Cut  out  a  circle  having  very  few 
sectors;  arrange  them  in  a  straight 
line.  Cut  out  another  circle  of  the 
same  size  having  very  many  sectors; 
arrange  them  in  a  straight  line.  Com- 
pare the  two  groups  of  sectors. 

Now  study  this  figure  very  carefully. 
Can  you  see  the  radii  of  the  circle?  Can  you  see  the 
altitude  of  one  of  the  triangles?  Is  it  the  same?  Now 
study  the  circumference  of  the  circle  and  the  perimeter 
of  the  eight-sided  figure  or  octagon.  You  can  see  that 
there  must  be  a  httle  difference  between  the  lengths  of 
these,  but  supposing  we  used  a  sixteen-sided  figure  in 
place  of  an  eight-sided  figure  then  the  perimeter  would 
be  much  more  nearly  the  same  as  the  circumference  of 
the  circle,  and  if  we  used  a  figure  with  ever-so-many 
small  sides  there  would  be  practically  no  difference 
whatever,  and  the  area  of  all  the  Uttle  triangles  would 
then  be  practically  the  same  as  the  area  of  the  circle. 

As  a  matter  of  fact,  that  is  exactly  how^  we  find  the 
area  of  a  circle,  only  we  make  a  rectangle  out  of  the 
little  triangles  by  fitting  them  together  as  shown  here 


(VII-71) 


ARITHMETIC 

and  we  consider  ^  of  the  circumference  of  the  circle  as 
the  base  of  this  rectangle  and  the  radius  of  the  circle 
as  the  altitude  of  this  rectangle  which  represents  the 
area  of  the  circle. 

Since  a  circle  with  a  diameter  of  10  inches  has  a 
circumference  of  31.416  in.  and  a  radius  of  5  in.,  there- 
fore, 

31.416  sq.  in.  X  5       ^^  ^.        .  /.    •    i 
=  /8.54  sq.  m.,  area  of  circle. 

Your  short  rule  or  equation  for  finding  the  area  of 

a  circle  therefore  is: 

C  X  R       ,  C  X  D       , 

— 7i —  =  Area;  or  — -. =  Area. 

2  '  4 

Exercise  39 — Oral. 

1.  How  do  we  find  the  area  of  a  rectangle  whose  base 

is  3.1416  in.  and  whose  altitude  is  J  in.? 

2.  If  a  circle  has  a  circumference  of  3.1416  in.  and  a 

diameter  of  1  in.,  what  is  the  length  of  its  radius? 

3.  In  finding  the  area  of  a  circle,  we  think  of  the 

circumference  of  the  circle  as  being  what  dimen- 
sion of  a  rectangle? 

4.  In  finding  the  area  of  a  circle,  we  think  cf  the 

radius  of  the  circle  as  being  what  dimension  of 
a  rectangle? 

5.  How  do  we  find  the  area  of  a  circle  if  we  know 

the  radius  and  the  circumference? 

6.  What  must  we  get  first  or  think  first  if  we  know 

the  diameter  and  the  circumference  and  want 
to  find  the  area?     R  is  what  part  of  D?    Then 


^X  2  -^  X  ^-. 


(VII-72) 


MENSURATION 

7.  How  do  we  find  the  area  of  a  circle  if  we  know 

only  the  radius? 

8.  How  do  we  find  the  area  of  a  circle  if  we  know  only 

the  diameter? 

9.  How  do  we  find  the  area  of  a  circle  if  we  know 

only  the  circumference? 


Exercise  40 — Written. 

Find  the  area  of  the  following  circles,  using  the 
simplest  method  in  each  case: 

1.  R  =  4ft.;  C  =  25.1328ft. 

2.  D  =  10  ft.;  C  =  31.416  ft. 

3.  R  =  6  in. 

4.  C  =  314.16  yd. 

5.  D  =  i  in. 

6.  A  phonograph  record  has  a  diameter  of  10  in.; 

what  is  its  area? 

7.  A  circular  flower  bed  has  a  diameter  of  15  ft.; 

what  is  its  area? 

8.  What  is  the  diameter  of  the  largest  circle  which 

can  be  cut  out  of  a  14-in.  square?    What  is  the 
area  of  this  circle  using  3t  for  tt? 

9.  What  is  the  area  of  the  square? 

WTiat  is  the  difference  be- 
tween the  area  of  the  circle 
and  the  area  of  the  square 
out  cf  which  it  was  cut? 
What  is  the  ratio  of  the  area 
of  the  circle  to  the  area  of  the 
square?  Make  a  new  rule  for  finding  area.  If  D 
is  given,  this  would  be  a  fine  rule  to  use. 
(VII-73) 


1-4" 


ARITHMETIC 


10.  A  bull's  eye  target  has  two  sec- 
tions; the  inner  section  being 
white,  and  the  outer  section 
being  black.  If  the  diameter 
of  the  entire  target  is  4  ft., 
what  is  the  area? 

11.  If  the  diameter  of  the  white  section  of  this  target 

is  2  ft.,  what  is  its  area? 

12.  How  can  you  find  the  area  of  the  black  section? 

What  is  the   area   of    the 
black  section  of  the  target? 

13.  A  circular  cement  walk  2  yd. 

wide  runs  around  a  flower 

bed;   if  the  diameter  of  the 

flower  bed  is  6  yd.,  what  is 

the    area    of    the     cement 

walk? 

14.  A  semi-circular  paper  pattern  has  a 
diameter  of  8  in.;  what  is  the 
area  of  this  pattern? 

15.  A  quadrant  has  a  radius  of  2  ft. ;  what  is 

its  area? 

16.  What  is  the  area  of  the  bottom  of  a 

round  tin  pail,  the  diameter  being  14"? 
3t  in  cases  of  this  kind.) 

17.  What  is  the  area  of  the  bottom  of  a  drinking-cup, 

the  radius  being  If"?      (Use  31  in  cases  of  this 
kind.) 

18.  What  is  the  area  of  a  coin  having  a  diameter  of 

1"?      (Can  you  use  tt  of  the  square  of  the 
diameter  in  this  case?) 
CVII-74) 


(Use 


MENSURATION 


Lesson  19 

Finding  the  Area  of  the  Surface  of  a  Right 
(Rectangular)  Prism 


>— 


Z. 


I" 


I' 


2" 


=  e'LONG 


jr,      r-i 


Figure  I 


Fieure  TI 


Make  a  prism  out  of  heavy  paper. 

Besides  being  able  to  find  the  volume  of  solids,  we 
must  also  be  able  to  find  the  area  of  their  surfaces,  for 
it  is  only  in  this  way  that  we  can  tell  how  much  lumber 
is  required  to  construct  a  packing  case,  how  much  card- 
board is  needed  to  make  a  shoe  box,  etc. 

In  the  prism  shown,  the  surface  around  is  a  rectangle 
6''  long  by  4"  high,  containing  24  sq.  in. 

The  two  bases  are  rectangles  2'^  long  V  wide,  each 
containing  2  sq.  in.  or  a  total  of  4  sq.  in. 

The  entire  surface  therefore  contains  24  sq.  in.  +  4 
sq.  in.,  or  28  sq.  in. 

(VII-75) 


ARITHMETIC 

Exercise  41 — Written. 

Make  drawings  of  the  surfaces  of  the  following  rec- 
tangular prisms  (to  scale  of  I"  to  1",  or  |"  to  I'O  in  the 
same  way  as  is  shown  in  Figure  I,  and  find  the  area 
of  the  entire  surface  of  each: 

1.  3"  X  r  X  10'';  4.  r  X  2'  X  2'; 

2.  3'  X  6'  X    6';  5.  2  yd.  X  4  yd.  X  8  yd.; 

3.  r  X  V  X    4";  6.  V  X  2"  X  4". 

7.  How  many  square  inches  of  cardboard  are  needed 

to  make  a  box  for  an  umbrella,  the  size  of  the 
box  being  4''  wide,  4''  deep,  4'  high,  if  no  allow- 
ances are  made? 

8.  How  many  square  feet  of  zinc  are  needed  to  line 

a  case  2'  deep,  3'  wide,  4'  long? 

9.  How  many  square  yards  of  paper  are  needed  to 

hne  a  case  4'  X  4'  X  4'? 

10.  How  many  board  feet  of  1"  lumber  are  needed  to 

make  a  box  with  a  cover,  the  outside  dimensions 
being  20''  X  30"  X  40"? 

11.  Find  the  area  of  the  entire  surface  of  a  prism 

r  X  6"  X  8". 

12.  Find  the  area  of  the  entire  surface  of  a  cube  8§" 

each  way. 

13.  Find  the  area  of  the  entire  surface  of  a  cube 

having  5"  edges. 

14.  Find  the  area  of  the  entire  surface  of  a  prism 

r  X  4"  X  6". 

15.  Find  the  area  of  the  entire  surface  of  a  cigar  box 

r  X  6"  X  8''. 

16.  Find  the  area  of  the  four  walls,  floor,  and  ceiling 

of  a  room  14'  X  22'  X  10'. 
(VII-76) 


MENSURATION 


Lesson  20 
Finding  the  Area  of  the  Surface  of  a  Cylinder 


^HH^iE^ 


vO 


Make  a  cylinder  out  of  heavy  paper. 

A  ^'cylinder"  is  a  solid  bounded  by  a  uniformly 
curved  side  and  two  parallel  circular  ends  or  bases  of 
equal  size.  Pipes,  tin  cans,  and  round  pencils  usually 
have  the  shape  of  cyUnders. 

To  find  the  area  of  the  entire  surface  of  a  cyHnder, 
we  must  find  the  sum  of  the  areas  of  the  curved  side 
and  the  two  circular  bases. 

You  aheady  know  how  to  find  the  areas  of  the  two 
circular  bases;  therefore,  the  only  new  point  for  you 
to  learn  is  how  to  find  the  area  of  the  curved  side.  Can 
you  tell  how?     (Use  the  terms  of  the  cylinder.) 

In  the  drawing  here  shown,  we  find  the  circumference 
of  the  base  to  be  11";  therefore,  the  curved  side  is  11" 
long  and  6"  wide,  its  area  being  11  sq.  in.  X  6  =  66 

(VII-77) 


ARITHMETIC 

sq.  in.;  the  area  of  each  base  is  |  of  11  sq.  in.  X 
(3|  -^  2)  =  9f  sq.  in.  or  19  J  sq.  in.  in  both  bases,  or 
66  sq.  in.  +  19 J  sq.  in.  =  85^  sq.  in.  in  the  area  of 
the  entire  surface. 

Exercise  42 — Written. 

1.  Cut  a  paper  into  an  oblong  4"  X  8",  and  form  a 

hollow  cylinder  8''  long;  what  is  the  circumfer- 
ence of  this  cylinder?    Watch  the  bases  carefully. 

2.  Wliat  is  the  area  of  the  curved  side  of  this  cylinder? 

3.  What  is  the  diameter  of  one  of  the  bases  of  this 

cylinder? 

4.  What  is  the  area  of  each  of  the  two  bases? 

5.  What  is  the  total  area  of  the  curved  side  and  the 

two  bases? 

6.  A  section  of  rain  pipe  is  6"  in  diameter  and  36" 

long;  what  are  the  dimensions  of  a  sheet  of 
galvanized  iron  of  the  correct  size  to  make  this 
pipe,  allowing  |"  for  the  seam? 

7.  What  is  the  area  of  this  sheet  of  iron? 

8.  A  cylindrical  smoke-stack  is  12'  high  and  1'  in 

diameter;  what  is  the  area  of  its  curved  side? 
How  many  square  feet  of  metal  were  needed  to 
make  this  stack,  if  the  metal  overlaps  3''  at  the 
seam? 

9.  What  is  the  area  of  the  curved  side  of  a  pencil  7'^ 

long  and  f ''  in  diameter? 
10.  A  mailing  tube  is  20"  long  and  3"  in  diameter ; 
what  is  the  area  of  its  curved  side?     How  many 
such  tubes  could  be  cut  from  a  sheet  of  card- 
board 20"  by  37.7",  allowing  nothing  for  the 

seam? 

(VII-7S) 


MENSURATION 

Exercise  43 — Oral. 

1.  A  rectangular  prism  has  how  many  sides?     How 

many  bases?     How  many  surfaces? 

2.  How  do  we  find  the  area  of  the  entire  surface  of 

a  rectangular  prism? 

3.  Name  the  dimensions  of  each  of  the  surfaces  of  a 

prism  2"  X  4"  X  8". 

4.  How  many  sides  has  a  cylinder?      How  many 

bases?     How  many  surfaces? 

5.  What  is  the  shape  of  the  side  of  a  cylinder  when 

the  curve  is  straightened? 

6.  What  is  the  shape  of  each  of  the  bases  of  a 

cylinder? 

7.  Take  a  cylinder  and  tell  how  we  find  the  dimen- 

sions of  the  curved  side. 

8.  How  do  we  find  the  area  of  the  curved  side  of  a 

cylinder? 

9.  How  do  we  find  the  area  of  each  of  the  two  bases 

of  a  cylinder? 

10.  Take  a  cylinder  and  tell  the  class  all  you  can  about 

it.     Take  2  minutes. 

11.  Take  a  prism  and  tell  the  class  all  you  can  about 

it.     Take  2  minutes. 

12.  A  rectangular  prism  has  how  many  edges? 

13.  The  dimensions  of  a  rectangular  prism  are  5"  X 

10"  X  15";  how  many  of  its  edges  are  5"  long? 
How  many  of  its  edges  are  10"  long?  How 
many  of  its  edges  are  15"  long? 

14.  A  cylinder  has  how  many  edges? 

15.  The  diameter  of  a  cylinder  is  10";  what  is  the 

length  of  each  of  its  edges? 

(VII-79) 


ARITHMETIC 


Exercise  44 — Written. 

Problems  on  prisms  and  cylinders. 

(Use  Common  Fractions.) 


1. 


2"  Convex  surface  =  ? 


2.  Entire     surface 
=  ? 


3.    l: 


^-r/s!-^^ 


Entire  surface  =  ? 


-4.0' 


6.    2^: ^ 


How  many 
like  A 
can  you 
find  in 
B? 


J 


Entire  surface  =  ? 


6" 


Ei 


(VII-80) 


MENSURATION 


Lesson  21 
Cutting  Material  to  Avoid  Waste 


EXAMPLE:     How  many  sheets  6"  by 
10"  can  be  cut  from  a 
sheet  of  paper  20"  by   ^ 
36"?  e^» 


_6_  X      _2_ 
6j36        10)20 


=  12  sheets. 


6 '   6"  6"   6 "    6"   6' 

g 
b 


36" 


EXAMPLE:     How  many  cards  3"  by 
5"  can  be  cut  from  a 
sheet  16"  by  18"? 
3  X    _6       =18  cards. 
oTTe     3)18 

(1"  by  18"  remainder) 


yo 


3" 

3" 

3" 

3" 

3" 

3" 

10 

10 

10 

-I 

18" 


Material  such  as  paper,  tin,  cloth,  etc.,  often  comes 
in  large  sheets  or  pieces  out  of  which  smaller  sheets  or 
pieces  of  certain  sizes  are  to  be  cut.  We  have  paper 
cutting  for  our  work;  let  us  see  if  we  are  economical. 

While  it  is  alw^ays  possible  to  cut  a  sheet  having  an 
area  of  48  sq.  in.  into  two  sheets  each  having  an  area 
of  24  sq.  in.,  it  is  not  possible  to  obtain  two  sheets 
having  dimensions  of  6''  by  4''  unless  the  original  sheet 
has  dimensions  of  12''  by  4'^  or  dimensions  of  6'^  by  8"; 
therefore,  great  savings  are  made  by  using  material  of 
the  proper  size. 

To  tell  quickly  how  many  rectangular  sheets  of  a 
certain  size  can  be  cut  from  a  rectangular  sheet  of 
another  size,  and  also  to  determine  how  much  waste 

(vn-81) 


ARITHMETIC 

there  will  be,  we  divide  one  of  the  dimensions  of  the 
large  sheet  by  one  of  the  dimensions  of  the  small  sheet 
to  see  how  many  it  will  make,  then  we  divide  the  other 
dimension  of  the  large  sheet  by  the  other  dimension 
of  the  small  sheet  {in  each  case  using  as  the  divisor,  that 
dimension  which  will  leave  the  smallest  remainder)  ^ 
and  then  we  find  the  product  of  these  two  quotients. 


EXAMPLE :     How  many  let- 
ter heads  SV'  by  11"  can  be 
cut  from  a  sheet  30^"  by 
34"? 
4  X         2      =  8 

8^)34       ll)30i 

(8^"  by  34"  rem.) 

1  X     3=3 

d>mh       11)34 

(1"  by  8^"  rem.) _ 

Total,  11  Letter 
Heads. 


30  ys. " 


11" 

fl" 

s;^" 

; 

^ 

. 

00 

— 

- 

^ 

00 

: 

; 

00 

. 

- 

^ 

^* 

03 

. 

— 

Sometimes  it  is  most  economical  to  cut  the  material 
so  that  there  will  be  a  strip  remaining  which  will  be 
wide  enough  to  be  used  in  the  opposite  direction. 

Exercise  45— Oral. 

1.  What  is  the  area  of  a  piece  of  tin  3"  X  8"? 

2.  What  is  the  area  of  a  piece  of  tin  2"  X  6"? 

3.  The    answer   to   Question    1    was   what?      The 

answer  to  Question  2  was  what?     How  many 
times  as  large  as  No.  2  is  No.  1? 
(\7I-82) 


MENSURATION 

4.  Can  we  cut  No.  1  into  two  pieces  of  the  dimen- 

sions of  No.  2?     Why? 

5.  What  dimensions  should  No.  1  have  in  order  that 

we  might  cut  it  into  tw^o  of  No.  2? 

6.  What  other  dunensions  might  No.  1  have  which 

would  also  enable  us  to  cut  it  into  two  of  No.  2? 

7.  Explain  how  you  can  find  out  how  many  5"  X  8" 

cards  can  be  cut  from  a  sheet  2V  X  25"  in  size? 

8.  How  can  you  tell  if  there  will  be  any  waste? 

9.  In  each  of  the  divisions,  how  can  you  tell  which 

dimension  of  the  small  sheet  should  be  used  as 
the  di\dsor? 
10.  How  can  we  sometimes  cut  the  material  to  better 
advantage  than  at  first  appears  possible? 

Exercise  46 — Written. 

1.  How  many  cards  4"  X  6"  can  be  cut  from  a  sheet 

18"  X  20"?      How  much  waste  will  there  be? 
Show  this  by  a  diagram  drawn  to  the  scale  of  J. 

2.  How  many  sheets  of  tin  8"  X  10"  can  be  cut  from 

a  sheet  18"  X  20"?    What  are  the  dimensions  of 
the  remainder  if  there  is  one? 

3.  How^  many  pieces  of  clcth  4"  square  can  be  cut 

from  a  piece  12"  X  14"?     What  is  the  area  of 
the  remainder  if  there  is  one? 

4.  How  many  letter  heads  8  J"  X  11"  can  be  cut  from 

a  sheet  of  paper  22"  X  34"?     What  is  the  area 
of  the  remainder  if  there  is  one? 

5.  How  many  note  heads  5  J"  X  8  J"  can  be  cut  from 

a  sheet  of  paper  34"  X  44"?     How  many  can  be 
cut  from  a  ream  of  such  paper? 

(VII-83) 


ARITHMETIC 

6.  How  many  circulars  6"  X  9|"  can  be  printed  at 

one  time  on  a  sheet  of  paper  24"  X  38'7  How 
many  square  inches  would  be  wasted?  How 
many  reams  of  24"  X  38''  paper  would  be  needed 
for  8,000  circulars? 

7.  The  front  and  the  back  covers  of  a  magazine  are 

printed  on  one  sheet  of  paper;  if  the  finished 
magazine  is  10  J''  X  14'',  what  is  the  size  of  the 
complete  cover  as  it  comes  off  the  press?  How 
many  complete  covers  can  be  printed  from  a 
sheet  42"  X  63'7 

8.  If  both  sides  of  a  sheet  of  paper  24"  X  38"  are 

printed  and  made  into  a  book  6"  X  9§"  in  size, 
how  many  pages  will  the  book  have? 

9.  How  many  letter  heads  8 J"  X  11"  can  be  cut 

from  a  sheet  30 i"  X  34"? 
10.  A  pan  of  candy  is  20"  X  30";    into  how  many 
pieces  |"  X  11"  can  it  be  cut? 

Lesson  22 


Finding  the  Volume  of  a  Cylinder 


(dZZ^ 


A  ''cylinder,"  as  you  know,  is  a 
solid  bounded  by  a  uniformly  curved 
side  and  two  parallel  circular  ends  or 
bases  of  equal  size. 

As  in  the  case  of  the  prism,  we  find 
the  volume  of  a  cylinder  by  finding 
how  many  cubic  units  there  are  in 
each  la^^er  of  the  cylinder,  and  multi- 
plying this  by  the  number  of  layers. 
Therefore,  we  find  the  area  of  one 
(VII-84) 


MENSURATION 

of  the  bases,  make  that  one  unit  high  so  that  we  may 
know  how  many  cubic  units  there  are  in  each  layer, 
and  multiply  by  the  altitude  as  that  shows  the  number 
of  layers,  to  find  the  volume. 


EXAMPLE:  Find  the  volume  of  a  cylinder  whose  altitude  is 
10  in.,  the  diameter  of  the  base  being  4  in. 
The  area  of  the  base  =  12.5664  sq.  in.;  if  1  in.  high,  there 
are  12.5664  cu.  in.  in  each  layer;  as  the  height  is  10  in.,  there 
are  10  layers;  therefore,  the  volume  =  12.5664  cu.  in.  X  10, 
or  125.664  cu.  in.,  Ans. 


Exercise  47 — Oral. 

1.  Volumes  call  for  how  many  dimensions? 

2.  Volumes  call  for  the  use  of  what  table? 

3.  The  altitude  of  a  prism  is  6  in.  and  its  base  has 

an  area  of  10  sq.  in.;    how 

many  cubic  inches  are  there 

in  each  layer?      How  many 

layers  are  there?   How  many 

cubic  inches  are  there  in  the  prism?      What  is 

the  volume  of  this  prism? 

4.  The  altitude  of  a  cylinder  is  6  in.  and  its  base  has 
an  area  of  10  sq.  in. ;  how  many  cubic 
inches  are  there  in  each  layer?  How 
many  layers  are  there?  How  many 
cubic  inches  are  there  in  the  cylinder? 

What  is  the  volume  of  this  cylinder? 

5.  How  many  cubic  inches  are  there  in  each  layer  of 
a  cylinder  if  its  base  has  an  area  of  5  sq.  in.? 
How  many  layers  are  there  if  the  altitude  of  the 
cylinder  is  10  in.?     What  is  the  volume? 

(VII-85) 


ARITHMETIC 


6.  Name  some  prisms  that  you  see  or  know  about. 

7.  Name  some  cylinders  that  you  see  or  know  about. 

8.  How  do  we  find  the  volume  of  a  prism? 

9.  How  do  we  find  the  volume  of  a  cylinder? 

10.  Talk  about  a  prism  for  one  minute. 

11.  Talk  about  a  cylinder  for  one  minute. 

12.  How  do  you  find  the  area  of  the  base  of  a  cylinder 

if  you  know  the  diameter? 

Exercise  48 — Written. 

Find  the  volume  of  the  following  cylinders: 

1.  Area  of  base  =  14.5  sq.  in.;  Altitude  =  1  ft. 

2.  Area  of  base  =  45  sq.  yd. ;  Altitude  =  21  ft. 

3.  Area  of  base  =  23  sq.  ft.;  Altitude  =  2  yd. 

4.  An  oil  tank  is  If  ft.  in  diameter  and  4  ft.  high; 

how  many  gallons  of  oil  will  it  hold  if  there  are 
7 J  gallons  in  each  cubic  foot? 

5.  A   cylindrical   fish   tank   or  aquarium  is   12"  in 

diameter  and  10''  high;    how  many  gallons  of 
water  does  it  contain  when  completely  filled? 

K \2^ >j  6.  The  tank  here  shown 

half 


was  nait  full  of 
water  when  a  stone 
was  placed  in  the 
bottom  of  it,  mak- 
ing the  water  rise 
2";  what  was  the 
volume  of  the 
stone? 

(Note:   The  volume  of  any  insoluble  solid  object  of 
any  shape  can  be  measured  by  the  volume  of  water  it 

(VII-86) 


MENSURATION 

displaces.  In  this  case  the  stone  has  a  volume  equal 
to  a  cylinder  12"  in  diameter  and  2"  high,  since  it  made 
a  cylindrical  body  of  water  12"  in  diameter  rise  2".) 

7.  If  a  tank  had  a  diameter  of  4  ft.,  and  placing  a 

stone  in  it  made  the  water  rise  6  in.,  what  would 
be  the  volume  of  the  stone?     Draw  carefully. 

8.  How  many  cubic  feet  of  gas  must  enter  a  gas  tank 

200  ft.  in  diameter,  to  make  it  rise  from  a  height 
of  10  ft.  to  a  height  of  50  ft.?     Draw  carefully. 

9.  A  silo  is  40  ft.  in  diameter  and  80  ft.  high;   what 

is  its  volume  in  cubic  yards? 
10.  A  cylindrical  cistern  is  10  ft.  deep  and  4  ft.  in 
diameter;  how  many  gallons  of  water  will  it  hold? 

Exercise  49 — Oral  Review. 

1.  How  many  sections  are  there  in  a  township?    How 

many  square  miles?     What  are  the  dimensions 
of  a  township? 

2.  If  a  bricklayer's  wages  are  $30.00  for  a  week  of 

48  hours,  how  much  will  he  receive  for  working 
24  hours? 

3.  If  a  plasterer's  wages  are  $24.00  for  a  week  of  48 

hours,  how  much  will  he  receive  for  working 
30  hours? 

4.  What  is  the  interest  on  $400.00  for  90  days  at  6%? 

5.  What  is  the  ratio  of  the  circumference  of  a  circle 

to  the  diameter? 

6.  If  the  diameter  of  a  circle  is  10",  what  is  the  cir- 

cumference? 

7.  If  the  dimensions  of  a  box  are  3'  X  4'  X  5',  what 

are  the  dimensions  of  each  of  the  6  surfaces? 

(VII-87) 


ARITHMETIC 

8.  What  is  the  perimeter  of  a  4''  square?     What  is 

the  area? 

9.  Multiply: 

(«)  (h)  (c) 

60  X  12i;  56  X  25;  33  X  33i 

10.  Divide: 

(a)  (h)  (c) 

30 -^33 J;  800^161;  12 -- 25. 

(Time  for  #9  and  #10  should  be  3  minutes  or  less.) 

Exercise  50 — Written  Review. 

1.  What  is  the  interest  on  $438.00  from  March  8, 

1920,  to  May  22,  1920,  at  5%? 

2.  When  a  case  containing  3  reams  of  paper  was 

opened,  465  sheets  were  found  to  be  damaged; 
what  percentage  of  the  paper  was  undamaged? 

3.  Stating  your  answ^er  in  sq.  yd.,  sq.  ft.,  and  sq.  in., 

what  is  the  area  of  a  circle  having  a  circumfer- 
ence of  942.48  in.? 

4.  What  is  the  volume  of  a  section  of  water  main  4' 

in  diameter  and  10'  long? 

5.  What  is  the  area  of  the  curved  surface  of  10  sec- 

tions of  this  water  main? 

6.  What  is   the   volume   of  a   box  V  X  6"  X  18"? 

How  many  2'^  cubes  will  it  hold? 

7.  What  is  the  total  area  of  all  the  surfaces  of  this 

box?     (Answer  in  square  inches.) 
S.  If  this  box  is  made  of  wood  Y  thick,  and  a  cubic 
foot  of  this  wood  weighs  54#,  what  is  the  weight 
of  this  box? 

(VII-88) 


MENSURATION 

Add,  but  do  not  copy: 
(Time  for  these  5  examples  is  less  than  4|  minutes.) 


9. 

10. 

11. 

12. 

13. 

83,062 

2,763 

1,845 

1,972 

4,638 

912 

38,745 

4,618 

219 

7,319 

8,746 

428 

7,594 

35,211 

8,242 

29,428 

36 

8,730 

25 

3,625 

35 

46,812 

3,948 

740 

1,745 

7,318 

7,429 

9,378 

4,583 

9,721 
8,631 

4,728 
9,386 
5,581 
7,003 

1,784 

Copy  and  divide: 

(Time  for  these  6  examples  is  less  than  4  J 

minutes.) 

14. 

27,218  ^  62; 

17. 

29,580 

+  87; 

15. 

22,742  --  83; 

18. 

29,029 

+  91; 

16. 

31,434  --  39; 

19. 

37,666 

+  74. 

Copy  and  multiply: 

(Time  for  these  6  examples  is  less  than  4  J 

minutes.) 

20. 

434  X  24; 

23. 

6,831  X  123; 

21. 

639  X  53; 

24. 

3,805  X  416; 

22. 

728  X  62; 

25. 

7,312  X  548. 

Subtract,  but  do  not  copy: 
(Time  for  these  12  examples  is  less  than  4 J  minutes.) 
26.  27.  28.  29. 

316,817  943,814  740,002  800,005 

284,928  178,939  319,846  300,009 


(VIi-89) 


ARITHMETIC 

30. 

31. 

32. 

33. 

386,412 

745,038 

713,845 

385,219 

217,058 

296,839 

296,829 

300,951 

34. 

35. 

36. 

37 

731,987 

431,875 

589,463 

386,411 

645,879 

223,819 

331,861 

17,429 

(V1I-9C) 


PERCENTAGE 

Lesson  23 
Successive  Trade  Discounts 

In  your  previous  work  you  have  learned  that  ^Hrade 
discount"  is  an  amount  allowed  by  wholesalers  to 
retailers  so  that  retailers  can  sell  at  list  or  advertised 
prices  and  still  make  a  profit. 

As  some  merchants  allow  more  than  one  trade  dis- 
count, Ave  can  have  several  discounts  to  be  deducted 
one  after  the  other;  these  are  called  ''successive  dis- 
counts." 

In  deducting  successive  discounts,  each  discount  is 
figured  on  the  net  amount  remaining  after  deducting 
the  previous  discount;  therefore,  a  discount  of  10% 
and  10%  is  not  the  same  as  a  discount  of  20%,  because 
100%  less  10%  =  90%  and  90%  less  10%  of  itself  = 
81%,  while  100%  less  20%  =  80%. 

(>100%  C>-100% 

Less  10%  of^  10%  Less  20%  oP  20%^ 

Ct90%  80% 

Less  10%  of^  9% 
81% 
Again,  discounts  of  10%  and  5%  do  not  equal  15%. 
c>100%  ctlOO% 

Less  10%,  of^  10%  Less  15%,  of   15% 

^90%  85% 

Less    5%  of^  4i% 
85i% 

(VlI-91) 


ARITHMETIC 


EXAMPLE: 

$180.00  Less  20%,  10%,  and  5%  = 
$180.00 
36.00 


Less  20% 
Less  10% 
Less    5% 


144.00 

14.40 

129.60 

6.48 

$123.12 


EXAMI?LE: 

$180.00  Less  5%,  10%,  and  20%  =  ? 
$180.00 


Less  5% 
Less  10% 
Less  20% 


9.00 


171.00 
17.10 

153.90 

30.78 

$123.12 


Successive  discounts  may  be  deducted  in  any  order 
without  affecting  the  final  result;  thus,  a  discount  of 
20%,  10%,  and  5%  is  the  same  as  a  discount  of  5%, 
10%,  and  20%,  etc. 

The  difference  between  the  original  or  ''gross"  price 
and  the  ''net"  price  is  the  discount. 


EXAMPLE 

:  $180.00  Less  20*%,  10%,  and  5%  =  ? 

100% 

Less  20% 

20% 

80% 

Less  10% 

8%          68.4%  of  $180.00  =  $123.12; 

72% 

Less    5% 

3.6% 

68.4%, 

From  this  it  can  be  seen  that  taking  68.4%  of  any  amount 

gives  the  same  result  as  deducting  20%,  10%,  and  5%. 

Another  method  of  figuring  successive  discounts, 
which  saves  considerable  time  when  the  same  set  of 
discounts  is  to  be  deducted  from  several  amounts,  is  to 

(VII-92) 


PERCENTAGE 

find  what  per  cent  of  100%  remains  after  the  discounts 
are  deducted  and  then  find  this  percentage  of  the  various 
list  prices. 

Exercise  51 — Oral. 

1.  Is  a  trade  discount  of  20%  and  10%  the  same 

as  a  trade  discount  of  30%? 

2.  Explain  why. 

3.  Is  a  trade  discount  of  20%  and  10%  the  same  as 

a  trade  discount  of  10%  and  20%? 

4.  What  is  the  general  rule  regarding  the  order  in 

which  trade  discounts  may  be  deducted? 

5.  If  you  are  asbed  to  find  the  net  cost  of  an  article 

listed  at  S150.00  the  discount  being  10%  and 
10%,  state  how  you  would  proceed. 

6.  What  is  the  gross  price  of  an  article? 

7.  What  is  the  net  price  of  an  article? 

8.  What  is  the  difference  between  the  gross  price 

and  the  net  price  of  an  article? 

9.  What  is  the  easiest  w^ay  of  finding  successive  dis- 

counts when  the  same  set  of  discounts  is  to  be 
deducted  from  several  amounts? 

10.  What  per  cent  of  100%  remains  when  a  discount 

of  10%  and  10%  has  been  deducted? 

11.  If  you  w^ere  asked  to  deduct  10%,  20%,  and  5% 

from  six  different  amounts,  how  would  you  pro- 
ceed to  find  the  total  discount? 

12.  When  is  it  easier  to  find  the  per  cent  of  the  gross 

price  which  represents  the  net  price?  When  is 
it  easier  to  deduct  the  successive  discounts  one 
at  a  time? 

(VII-93) 


ARITHMETIC 

Exercise  52 — Written. 

Find  the  trade  discount  and   the   net  amount    of 
each  of  the  following  bills  and  prove  your  work: 

1.  $140.00  less  10%  and  5%; 

2.  $80.00  less  15%  and  10%; 

3.  $200.00  less  10%  and  10%; 

4.  $450.00  less  20%  and  10%. 

5.  A  wholesale  grocer  allows  a  discount  of  20%  and 

10%  from  list  prices  on  soap;  what  is  the  net 
price  of  a  box  containing  150  cakes  of  soap  listed 
at  10^  per  cake? 

6.  A  hat  manufacturer  allows  a  trade  discount  of 

20%  and  10%  on  his  $3.00  hats;  what  is  the 
net  cost  of  5  dozen  of  such  hats? 

7.  What  per  cent  of  discount  is  equivalent  to  succes- 

sive discounts  of  20%,  10%,  and  5%? 

8.  If  a  music  publisher  allows  a  discount  of  40% 

and  10%  from  list  prices,  what  would  be  the 
net  amount  he  would  charge  for  each  of  the 
following  items : 

10  Pieces  of  Sheet  Music @  50^ 

20  Pieces  of  Sheet  Music @  75^ 

A2  Music  Books @  $1.50 

8  Music  Collections @  $2.00 

24  Songs @25f^ 

12  Duets @  $1.25 

9.  What  is  the  discount  on  $125.00  at  the  rate  of 

20%,  20%,  and  10%? 
10.  At  the  rate  of  25%,  10%,  and  10%,  what  is  the 
discount  on  $160.00? 

(VII-94) 


PERCENTAGE 

Lesson  24 

Finding  the  Gross  Amount  When  the  Rates  of 
Discount  and  the  Net  Amount  Are  Given 


EXAMPLE  :^After  deducting  10% 

and    10%,  the 

net  "amount 

of  an   invoice   is  S324.00;      what 

is  the   gross 

amount  of  the  invoice? 

$324.00 

=  100%  less  10%  and  10%; 

100% 

therefore, 

-  10%  (10% 

of  100%) 

$324.00 

=  81  %  of  the  Gross  Amount 

90% 

$324.00 

4-81%  =  $400.00,  Ans. 

-   9%  (10%  of  90%) 
81%  Net  Rate  Per  Cent 

As  you  have  already  learned,  the  gross  amount 
always  is  100%;  therefore,  when  the  net  amount  and 
the  rates  of  discount  are  known,  we  must  subtract  the 
rates  of  discount  successively  from  100%  so  that  we 
may  know  what  per  cent  the  net  amount  is  of  the 
gross  amount,  and  then  we  must  divide  the  net 
amount  by  this  per  cent  to  find  100%  or  the  original 
bill. 

Exercise  53 — Oral. 

Give  the  total  discount  rate  per  cent,  also  the  net 
rate  per  cent  of  the  following: 

1.  10%  and  10%;  7.     5%  and  20%; 

2.  20%  and  10%?;  8-     5%  and  10%; 

3.  10%)  and  20%^;  9.  25%  and  10%; 

4.  10%oand    5%;  10.  25%  and    5%; 

5.  25%  and  20%;  11.  20%  and  20%; 

6.  20%o  and    5%;  12.  30%  and  10%. 

(VII-95) 


ARITHMETIC 

13.  If  the  net  amount  of  an  invoice  is  $80.00  after 

deducting  20%  and  20%,  what  is  the  first  thing 
you  would  do  if  you  were  asked  to  find  the  gross 
amount? 

14.  Since  a  discount  of  20%  and  20%  is  the  same  as  a 

discount  of  36%,  what  per  cent  of  the  gross 
amount  is  $80.00  in  Question  13? 

15.  Knowing  that  $80.00  is  a  certain  per  cent  of  the 

gross  amount,  how  do  we  find  the  gross  amount? 

16.  If  10%  and  10%  discount  on  a  certain  invoice  is 

$19.00,  how  would  you  find  the  gross  amount 
of  the  invoice? 

17.  If  the  discount  on  a  certain  invoice  is  $43.00,  and 

the  net  amount  of  the  invoice  is  $88.00,  how 
would  you  find  the  gross  amount? 

18.  In    the    example   stated    in   Question    17,   how 

would  you  find  the  rate  of  discount? 

19.  In  Question  17,  how  would  you  find  what  per 

cent  the  net  amount  is  of  the  gross  amount? 

Exercise  54 — Written. 

Solve  and  prove: 

1.  After  deducting  20%  and  10%  discount,  the  net 

amount  of  an  invoice  is  $108.00;  w^hat  is  the 
gross  amount  of  the  invoice? 

2.  After  deducting  20%  and  10%,  the  net  amount 

of  an  invoice  is  $85.50;  what  is  the  gross 
amount  of  the  invoice? 

3.  A  clock  manufacturer  changed  his  trade  discount 

from  20%  to  10% ;  what  per  cent  did  he  increase 
his  net  price? 

(VII-96) 


PERCENTAGE 

4.  If  25%  and  10%  discount  on  an  invoice  is  $81.25, 

what  is  the  gross  amount  of  the  invoice? 

5.  If  20%  and  5%  discount  on  an  invoice  is  $57.60, 

what  is  the  net  amount  of  the  invoice? 

6.  If  25%  and  20%  discount  on  an  invoice  is  $64.00, 

what  is  the  gross  amount  of  the  invoice? 
Find  the  missing  items  in  the  following: 


Gross 

Amount 

Net 
Amount 

Rate  of  Discount 

Rate  Per  Cent 

Net  Amount  is 

of  Gross  Amount 

'7. 

$2,000. 

? 

20%  and  10% 

? 

8. 

$1,650. 

? 

10%  and  10% 

? 

9. 

S250. 

? 

10%  and    5% 

? 

10. 

9 

$526.50 

10%  and  10% 

? 

11. 

? 

$3,600. 

20%  and  10% 

? 

12. 

? 

$873. 

3%  and  10% 

? 

13. 

$5,000. 

? 

10%  and  10% 
and  10% 

? 

Lesson  25 
Insurance 


Exercise  55 — Oral. 


In  an  Insurance  Office. 
Class  is  to  select  two  girls  and  two  boys  to  be  the 
insurance  men  with  offices  in  opposite    sides  of   the 
room.     Some  of  the  pupils  are  to  be  ready  to  inquire 
about  insurance  for  fire. 

(VII-97) 


ARITHMETIC 


Choose  some 
of  these 
questions 
and  get  the 
appointed 
insurance 
men  in 
office  to 
answer  you. 


Class  mem- 
bers may 
stand  and 
call  for  a 
clearer  un- 
derstanding 
on  any 
point. 

Other 
pupils  are 
to  carry 
out  this 
part  of  the 
work.     Two 
or  three 
pupils  go 
as  members 
of  one 
family  to 
take  out 
insurance. 


1.  Questions  are  to  be  asked 

regarding  value,  rate, 
premium  and  amount  of 
insurance  possible: 

a)  For  frame  house; 

b)  For  brick  house; 

c)  For  large  apartment  build- 

ing; 

d)  For  a  store; 

e)  For  frame  house  next  to 

dry  cleaning  place; 
/)  For  doctor's  office; 
g)  For  a  dry  cleaning  place; 
h)  For  a  brand-new  house; 

2.  How  much  insurance  can 

you  get  or  do  you  want? 

3.  Who  tells  the  value  of  the 

place? 

4.  Who   decides   how   much 

you  can  get? 

5.  Is    the    rate    always    the 

same? 

6.  If  it  differs,  why? 

7.  Go  take  out  insurance  to 

cover  your  building. 

8.  Go  take  out  insurance  to 

cover  your  furniture. 

9.  Go  take  out  insurance  to 

cover  your  personal  prop- 
erty.    (Tell  what  it  is.) 

(VII-98) 


Appointed 

insurance 
men  in 
office  must 
answer. 

pet  all  the 
information 
you  can. 

Ask  father, 
mother, 
teacher,  and 
insurance 
men. 


PERCENTAGE 


Class  mem- 
bers may 
stand  and 
call  for  a 
clearer  un- 
derstanding 
on  any 
point. 

Other 
pupils  are 
to  carry 
out  this 
part  of  the 
work.     Two 
or  three 
pupils  go 
as  members 
of  one 
family  to 
take  out 
insurance 


10.  Ask  for  rates  for  1  yr.,  3 

yr.,  5  yr. 

11.  Choose. 

12.  Pay  whom  for  the  insur- 

ance?     (The   Insurance 
Company.) 

13.  What  is  that  money  called? 

(Premium.) 

14.  What  is  your  contract  or 

important  paper  called? 

15.  ^¥here  should  you  keep  it? 

Why? 

16.  What  is  the  ''face"  of  the 

poHcy? 


Exercise  56 — Oral. 

1.  What  is  the  premium  on  a  $2,000.  insurance  policy 

at  i%?        - 

2.  My  barn  is  worth  $800.     If  I  have  an  insurance 

policy  on  J  of  its  value  at  J%,  what  must  I  pay 
for  my  insurance? 

3.  The  Continental  Insurance  Company  insures  John 

Black's  furniture  against  loss  by  fire  to  the 
extent  of  $800.  at  90^  per  yr.  per  $100.  Who  is 
the  insured?  Who  is  the  insurer?  What  is  the 
written  agreement  called?  How  much  premium 
did  he  pay?  State  the  face  of  the  policy.  What 
kind  of  insurance  is  this? 

(Examine  an  insurance  policy  and  be  ready  to 
talk  about  it.) 

4.  Can  you  name  some  other  insurance? 

(Vn-99) 


ARITHMETIC 

5.  What  is  Life  Insurance?  Accident  Insurance? 
Marine  Insurance?  Employers'  Liability  Insur- 
ance? 

Exercise  57 — Written. 

1.  A  stock  of  merchandise  valued  at  $5,000.  is  insured 

at  45j^  per  $100.;  what  is  the  premium  on  this 
policy? 

2.  A  building  valued  at  $8,000.  is  insured  for  90% 

of  its  value  at  $1.20  per  $100.;  what  is  the 
premium? 

3.  A  vessel  worth  $200,000.  is  insured  for  80%  of  its 

value  at  4^%,  and  its  cargo,  worth  $450,000.,  is 
insured  for  90%  of  its  value  at  4f%;  what  is  the 
total  premium  on  the  two  policies? 

4.  A  man  aged  30,  insured  his  life  in  favor  of  his  wife 

for  $10,000.,  agreeing  to  pay  an  annual  premium 
of  $20.50  per  $1,000.  as  long  as  he  lived.  He 
died  at  the  age  of  50.  How  much  had  he  paid 
in  premium  during  the  20  yr.?  How  much  did 
his  wife  get  from  the  insurance  company? 

5.  If  a  premium  of  $80.00  was  paid  for  insurance  at 

i%  to  cover  I  of  the  value  of  my  house,  what 
was  the  value  of  my  house? 

Lesson  26 

Commission  and  Brokerage 

You  have  already  learned  that  ^^ commission"  is  an 
amount  of  money  which  is  paid  by  one  person  who  is 
called  the  ^'principal"  to  another  person  who  is  called 

(VII-IOC) 


PERCENTAG-E 

the  "agent/'  for  some  service  which  the  agent  performs 
for  the  principal. 

A  '^ commission  merchant"  is  one  who  sells  produce 
or  merchandise  of  a  similar  nature  for  his  principal; 
however,  while  an  ordinary  salesman  who  sells  on  com- 
mission assumes  no  responsibility  in  regard  to  collecting 
for  the  goods  he  sells,  a  commission  merchant  is  the 
Icser  if  he  fails  to  collect  for  the  merchandise  he  sells. 

A  shipment  of  merchandise  to  be  sold  by  a  conamis- 
sion  merchant  is  called  a  '^consignment,"  and  the 
amount  of  money  which  is  to  be  paid  to  the  principal 
after  commissions  and  all  other  charges  are  deducted 
is  called  the  ''net  proceeds." 

A  "broker"  also  sells  on  a  percentage  basis,  but  his 
business  is  usually  confined  to  real  estate,  stocks,  bonds, 
etc.  The  amount  he  receives  for  his  work  is  called  the 
"brokerage." 

Exercise  58 — Oral. 

Commission  and  Brokerage  Project 
Select  two  girls  to  be  real  estate  brokers,  and  two 
boys  on  the  opposite  side  of  the  room  to  be  coromission 
merchants. 

1.  (The  commission  merchants  are  to  impersonate 
Henderson  &  Co. ;  one  of  the  pupils  can  imper- 
sonate John  Smith,  the  shipper;  others  can 
impersonate  the  buyers;  others  can  ask  the 
questions  shown;  everyone  may  ask  for  infor- 
mation.) 

John  Smith,  who  owns  a  fruit  farm  in  Michi- 
gan, sent  100  crates  of  berries  to  Henderson  & 

(VII-lOl) 


ARITHMETIC 

Co.,  Commission  Merchants,  Chicago,  Ilhnois. 
The  berries  were  sold  for  $7.00  per  crate,  and  a 
commission  of  10%  was  charged.  The  freight 
on  the  shipment  amounted  to  $5.00. 

(a)  In  this  transaction,  who  is  the  principal? 

(6)   Who  is  the  agent? 

(c)  What  did  this  consignment  consist  of? 

(d)  What  amount  was  received  for  tlie  berries? 

(e)  What  amount  of  commission  was  charged? 
(/)  How  much  was  paid  to  the  principal  as  his  net 

proceeds  after  deducting  the  commission 
and  the  freight  charges? 
2.  (The  real  estate  brokers  are  to  impersonate 
Frederick  Sharp;  others  can  impersonate  the 
seller,  Henry  Wise;  others  can  impersonate  the 
buyer;  others  can  ask  the  questions  shown; 
everyone  may  ask  for  information.) 

A  real  estate  broker  named  Frederick  Sharp 
sold  a  farm  belonging  to  Henry  Wise  for  $10,000., 
receiving  5%  as  his  brokerage, 
(a)  In  this  transaction,  who  is  the  principal? 
(6)  Who  is  the  agent? 

(c)  Who  is  the  broker? 

(d)  How    much    brokerage    was    paid?       Who 
received  it? 

(e)  What  did  the  net  proceeds  amount  to?     Who 
received  them? 

Now  some  may  ask  these  questions  and  others  may 
answer;  the  real  estate  brokers  must  be  ready  to  settle 
any  point  about  brokerage ;  the  commission  merchants 
must  be  ready  to  settle  any  point  about  commission. 

(VII-102) 


PERCENTAGE 

3.  In  commission  and  brokerage,  who  is  the  principal? 

4.  Who  is  the  agent? 

5.  What  is  a  commission  merchant? 

6.  What  is  a  real  estate  broker? 

7.  What  is  the  amount  called  which  a  conamission 

merchant  receives  for  his  services?     Is  it  more 
or  less  than  100%? 

8.  What   is   the   amount   called  which   a  principal 

receives?     Is  it  more  or  less  than  100%? 

9.  What  is  the  amount  called  which  a  broker  receives 

for  his  services?     Is  it  more  or  less  than  100%? 
10.  State  whether  commission  or  brokerage  was  paid 
on    the    following    transactions,    and   give    the 
amount  so  paid: 

(a)  The  sale  of  merchandise  amounting  to  $500. 

at  6%; 

(b)  The  sale  of  fruit  amounting  to  $300.  at  12%; 

(c)  The  sale  of  produce  amounting  to  $800.  at  8% ; 

(d)  The  sale  of  a  house  and  lot  for  $8000.  at  5%; 

(e)  The  sale  of  Liberty  Bonds  for  $800.  at  |%. 

Exercise  59 — Written. 

Solve  and  prove: 

1.  A  commission  merchant  sold  the  following  consign- 
ment of  merchandise : 

20  boxes  Oranges @  $2.00 

24  boxes  Lemons @    3.00 

25  boxes  Grape  Fruit @    2.80 

12  bunches  Bananas @    1.50 

He  received  8J%  commission;  the  freight,  dray- 
age,  etc.,  amounted  to  $4.50;   what  amount  of 


ARITHMETIC 

commission  was  paid?  What  did  the  net  pro- 
ceeds amount  to? 

2.  What  is  the  commission  at  4%  on  50  baskets  of 

peaches  sold  at  60^  per  basket? 

3.  The  commission  on  the  sale  of  a  certain  lot  of 

merchandise  was  paid  at  the  rate  of  8%  and 
amounted  to  $32.00;  what  was  the  amount  of 
the  sale? 

4.  A  consignment  of  3,465  bu.  of  wheat  was  sold  at 

83 f^  per  bu.;  what  was  the  agent's  commission 
at  2i%? 

5.  A  house  and  lot  was  sold  for  $6,575.00:  the  broker- 

age was  4%,  and  the  attorney's  fees,  etc., 
amounted  to  $37.00;  what  were  the  net  pro- 
ceeds of  this  sale?  If  you  were  the  broker,  how 
would  you  send  this  money  to  the  principal? 
Why? 

6.  If  the  brokerage  on  the  sale  of  a  certain  farm  was 

paid  at  the  rate  of  3|%  and  amounted  to 
$140.00,  for  what  amount  was  the  farm  sold? 
How  many  acres  were  there  if  the  price  per  acre 
was  $50.00? 

7.  If  8%  conmiission  on  the  sale  of  a  consignment  of 

potatoes  amounts  to  $11.40,  and  the  potatoes 
were  sold  at  $4.75  per  bbl.,  how  many  barrels 
were  there  in  the  consignment? 

8.  The  net  proceeds  from  the  sale  of  a  consignment 

of  eggs  amounted  to  $38.40;  if  4%  commission 
was  charged  by  the  commission  merchant,  what 
amount  was  received  for  the  eggs? 

9.  The  net  proceeds  from  the  sale  of  a  consignment 

(VII-104) 


PERCENTAGE 

of  apples  amounted  to  S665.00;  if  5%  commis- 
sion was  charged,  and  the  apples  were  sold  for 
$1.40  per  bbl.,  how  many  barrels  were  there  in 
the  consignment? 
10.  The  net  proceeds  from  the  sale  of  a  factory 
site  amounted  to  $7,560.00;  if  the  brokerage 
amounted  to  $540.00,  what  was  the  rate  per 
cent  of  brokerage? 

Lesson  27 
Taxes 

A  ^Hax"  is  a  sum  of  money  which  must  be  paid  by 
the  citizens  to  help  defray  the  expenses  of  the  national, 
state,  county,  and  city  governments,  and  for  public 
schools,  improvements,  etc. 
Taxes  are  of  several  kinds : 

Real  estate   taxes   on   the   assessed  valuation  cf 

houses,  lots,  etc. 
Personal  property  taxes  on  the  assessed  valuation 

of  movable  property. 
Income  taxes  on  salary  and  other  income. 
^^ Assessed  valuation"  means  the  estimated  value  of 
the  property  which  is  subject  to  the  tax,  or  upon  which 
tax  is  paid,  as  determined  by  the  Assessor. 

Taxes  are  charged  on  a  percentage  basis,  but  the 
percentage  is  sometimes  stated  as  being  a  certain 
number  of  mills  per  dollar,  or  a  certain  number  of  cents 
per  hundred  dollars. 

A  certain  portion  of  a  person's  income  is  usually  not 
subject  to  income  taxes;  this  is  called  an  ''exemption." 

(yii-105) 


ARITHMETIC 

Exercise  60 — Oral. 

Tax  Project 

Select  an  Assessor;  a  Board  of  Review  for  the  adjust- 
ment of  complaints;  a  Tax  Office  Cashier. 

Let  the  Assessor  go  around  and  assess  the  real  and 
personal  property  of  the  property  owners;  the  teacher 
will  give  the  tax  rates  for  the  different  to\\Tiships  (differ- 
ent parts  of  the  class  room) ;  those  who  think  they  are 
assessed  too  high  can  go  before  the  Board  of  Review 
for  a  hearing;  others  must  pay  their  taxes  at  the 
Cashier's  Office;  some  may  ask  the  following  questions 
and  others  may  answer;  all  may  ask  for  information; 
the  Officials  must  be  ready  to  settle  disputes. 

1.  A  man  who  owns  a  house  and  lot  having  an 

assessed  valuation  of  $3,500.  has  to  pay  a  tax 
at  the  rate  of  2%. 
(a)  Name  some  of  the  purposes  for  which  this  tax 

may  be  used  when  collected. 
(6)  Is  this  a  real  estate  tax,  a  personal  property 

tax,  or  an  income  tax? 
(c)   What  is  meant  by   ^^  assessed  valuation  of 
$3,500."? 

2.  A  rich  man  has  personal  property  of  an  assessed 

valuation  of  $100,000.  and  lives  in  a  city  w^here 
the  tax  rate  is  $1.75  per  $100. 

(a)  What  kind  of  a  tax  does  this  man  pay? 

(b)  What  rate  per  cent  is  this  tax? 

(c)  Tell  how  to  find  the  amount  of  this  tax. 

(d)  What  is  the  amount  of  this  man's  tax? 

3.  Why  must  citizens  pay  taxes? 

4.  What  is  a  real  estate  tax? 

(VII-106) 


PERCENTAGE 

5.  What  is  a  personal  property  tax? 

6.  What  is  an  income  tax? 

7.  Name  three  kinds  of  income  that  a  man  may 

receive. 

8.  What  is  meant  by  '' assessed  valuation"? 

9.  What  is  meant  by  "exemption"? 

10.  What  is  a  Tax  Assessor? 

11.  What  is  a  Board  of  Review? 

Exercise  61 — Written. 

1.  If  the  tax  rate  is  2%,  what  is  the  tax  on  a  house 

and  lot  having  an  assessed  valuation  of  $12,500.? 

2.  If  the  tax  rate  is  13  mills  on  the  dollar,  what  is 

the  tax  on  a  farm  having  an  assessed  valuation 
of  $25,300.? 

3.  If  the  tax  rate  is  $1.25  per  $100.,  what  is  the  tax 

on  a  piece  of  property  having  an  assessed  valua- 
tion of  $9,540.? 

4.  If  the  assessed  valuation  placed  on  property  is 

33 J%  of  the  actual  value,  and  the  tax  rate  is 
5%,  what  would  be  the  tax  on  an  estate  having 
an  actual  value  of  $90,000.? 

5.  If  the  assessed  valuation  of  all  the  taxable  prop- 

erty in  a  town  is  $1,530,000.,  and  the  following 
table  shows  the  items  that  make  up  the  tax  rate, 
what  is  the  total  tax  to  be  collected? 

Per  $100. 

Schools $0,255         City $2.16 

Drainage 113         County 52 

Parks 085         State 11 

(YII-107) 


ARITHMETIC 

6.  What  rate  of  taxation  must  be  used  to  raise 

$10,942.50  in  a  town  having  a  total  assessed 
valuation  of  $875,400.? 

7.  In  a  certain   town    there    is  real  and    personal 

property  assessed  at  $3,245,000.;  if  the  rate  of 
taxation  is  such  that  a  total  of  $45,430.  will  be 
raised,  how  much  must  a  man  pay  who  has 
property  assessed  at  $10,025.? 

8.  If  a  tax  rate  of  18  mills  on  the  dollar  yields 

$43,869.60,  what  is  the  assessed  valuation  of 
the  taxable  property? 

9.  A  tax  collector's  commission  is  2  J%  on  the  amount 

of  taxes  collected.  What  will  be  his  commission 
if  the  assessed  value  of  the  property  is  $6,875,400. 
and  the  rate  of  taxation  is  $2.37^  per  $100.00? 

10.  A  certain  community  wishes  to  build  a  new  town- 

hall  to  cost  $48,500. ;  w^hat  rate  of  taxation  must 
be  used  if  the  assessed  valuation  of  the  property 
is  $2,000,000.  and  3%  of  the  amount  collected 
must  be  paid  to  the  collector? 

11.  How  much  income  tax  must  Mr.  Jones,  a  single 

man,  pay  if  his  income  is  $5,400.  during  a  certain 
year  and  he  is  entitled  to  an  exemption  of  $1,000., 
the  tax  rate  being  2%  on  the  first  $2,500.  of  tax- 
able income  and  3%  on  the  balance? 

Mr.  Tower's  Tax  Trouble 

12.  Mr.  Tower  has  a  married  man's  exemption  under 

certain  income  tax  laws  of  $2,000.  plus  $400.  for 
each  child  under  the  age  of  16  years.  How 
much  income  tax  must  he  pay  if  he  has  3  young 

(VII-IOS) 


PERCENTAGE 

children  and  his  income  is  $14,350.  during  a 
certain  year,  the  tax  rate  being  as  follows: 

2%  on  the  first  S2,500.  subject  to  tax; 

3%  on  the  next   2,500.  subject  to  tax; 

4%  on  the  next   2,500.  subject  to  tax; 

5%  on  the  next   2,500.  subject  to  tax; 

8%  on  the  next   2,500.  subject  to  tax; 
10%  on  the  next   2,500.  subject  to  tax. 
13.    The     following     year,     Mr.     Tower's     income 
amounted  to  $12,500.     Allowing  the  same  ex- 
emptions as  before,  how  much  tax  must  he  pay, 
the  tax  rates  being  as  follows: 

2%  on  the  first  $2,000.  subject  to  tax; 
2i%  on  the  next  $2,000.  subject  to  tax; 

3%  on  the  next  $2,000.  subject  to  tax; 
3i%  on  the  next  $2,000.  subject  to  tax; 

4%  on  all  over  $8,000.  subject  to  tax. 

Lesson  28 
Computing  Interest  When  There  Are  Partial  Payments 

You  already  know  that  '^interest"  is  money  paid, 
or  to  be  paid,  for  the  use  of  money. 

As  we  can  charge  interest  on  a  sum  for  only  such 
time  as  the  sum  remains  unpaid,  all  partial  payments 
on  the  principal  must  naturally  affect  the  amount  of 
interest;  therefore,  whenever  a  partial  payment  is 
made,  the  amount  of  the  partial  payment  must  be 
subtracted  from  the  principal  and  the  difference  so 
found  is  the  new  principal  to  the  date  of  the  next 
partial  payment. 

Partial  pajonents  should  be  endorsed  on  the  back  of 
the  note. 

(VII-109) 


ARITHMETIC 


EXAMPLE:  'What  is  the  amount  due  on  Oct.  10,  1924,  on  the 
following  note,  payments  having  been  made  as 
shown  by  the  endorsements  on  the  reverse  side: 

The  Endorsements 
on  the  Reverse 
The  Face  of  the  Note.  Side  of  the  Note. 


Chicago,  111.   ^)<Ar  /O.  /^t.3 
\^un  f^n.yrr^-cr^^^i^J^AU^x  date^£_promlse  to  pay 
to  the  order  of-iA''c;A/tx|  -^.  •Cr./<gt.,Aifey 


C^CtAAJi 


V^v 


^>^!L<r»t^z^   /^-^-<-A^ 


.Dollars 


Value  received  with  Interest  at^_%  per  a^num 


Received  on  Within  Note 


Explanation : 
Principal : 

Principal  10/10/23 S450.00 

Payment  2/10/24 50.00 

Prmcipal  2/10/24 400.00 

Payment  5/10/24 50.00 

Principal  5/10/24 350.00 

Payment  8/10/24 100.00 

Principal  8/10/24 250.00 

Interest : 

10/10/23  to    2/10/24  ($450.  4  mo.  @  6%)  $9.00 

2/10/24  to    5/10/24  ($400.  3  mo.  @  6%)    6.00 

5/10/24  to    8/10/24  ($350.  3  mo.  @  6%)    5.25 

8/10/24  to  10/10/24  ($250.  2  mo.  @  6%)    2^ 

Total  Interest $22.75 

Amount  due  10/10/24 $272.75,  Ans. 


(VII- 110) 


PERCENTAGE 

Exercise  62 — Oral. 

1.  What  are  the  dates  and  the  amounts  of  the  pay- 

ments on  the  note  shown  in  the  example? 

2.  Why  is  the  interest  from   10/10/23   to  2/10/24 

figured  on  $450.? 

3.  What  is  the  balance  of  the  principal  after  deduct- 

ing the  payment  of  2/10/24? 

4.  Why  is  the  $50.  payment  of  5/10/24  deducted 

from  the  $400.00  principal? 

5.  Explain  the  entries  made  on  8/10/24  and  state 

why  each  of  these  entries  was  made. 

6.  Explain  each  of  the  interest  calculations,  and  show 

why  each  is  figured  on  a  different  principal. 

7.  What  is  interest? 

8.  How  do  we  figure  the  interest  when  there  are 

partial  pajnnents? 

9.  If  someone  holds  your  note  for  $500.00  and  you 

pay  $100.00  en  it,  what  kind  of  a  receipt  should 
you  demand  for  the  payment? 
10.  Why  is  the  interest  on  any  sum  for  60  days  at 
6%  equal  to  1%  of  the  sum? 
What  is  the  interest  on  each  of  these  notes: 


Principal 

Time 

Rate 

11. 

$350. 

60  days 

6% 

12. 

$400. 

30  days 

6% 

13. 

$825. 

120  days 

6% 

14. 

$200. 

90  days 

6% 

15. 

$600. 

60  days 

5% 

16. 

$400. 

30  days 

3% 

17. 

$300. 

60  days 

4% 

18. 

$500. 

90  days 
(yil-ni) 

6% 

ARITHMETIC 

Exercise  63 — Written. 

What  is  the  date  of  maturity  and  the  amount  pay- 
able at  the  date  of  maturity  of  each  of  these  notes: 

1.  Principal,  $350.00;  Date,  Apr.  6,  1918;  Time,  4 

months;  Int.,  6%; 
Partial  payments:    May  6,  1918,  $100.00; 

July  6,  1918,  $100.00. 

2.  Principal,  $148.00;  Date,  Dec.  6,  1923;  Time,  90 

days;  Int.,  5%; 
Partial  payments:    Jan.  6,  1924,  $50.00; 

Feb.  6,  1924,  $50.00. 
3. 


^sy^n  -7^ 


New  York,_^ 19  J^ 

^/ af  teTdate.^_proniise  to  pay 


to  the  order  of      ^t^y-t..t;t^',.t^)'7^2^^^  

^TiLi^'/TT^ZTj^r^  <^'^-Au>i>L^(i£>tje-^  ^^^^^T- — ^ Dollars 

At  ^u:z^?^£:6^^        /^^2.^u<Je^ 

Value  T^c.^\^A.MrC€lu>^^UuAj^X^oM  s£'yo '^xaJ ^2m/ujj^^ 
No.  /OS       Due  /2Jx/,'^/^i^f\    ^:^4^/^ty  z^AjLyz-c^AAr 


Payments  Endorsed : 
April  8,  1920,  $1,200. 
June  8,  1920,  $1,000. 
What  amount  was  paid  October  20,  1920,  on  which 
date  the  note  was  settled  in  full? 

(VII-112) 


PERCENTAGE 

Lesson  29 

Finding  the  Principal  When  the  Time,  Rate,  and 
Interest  Are  Given 


■5!!nHB"!!f?^5B5? 


EXAMPLE:     What  principal  will  yield  $8.70  in  90  days  at  6%? 

Interest  on  $1.00  for  90  days  at  6%  =  W, 

$8.70  4-  $0,015  =  $580.00,  Ans. 

Proof:  Int.  on  $580.  for  90  days  at  6%  =  $8.70. 

I  -  (T  X  R)  =  P 

T  X  R  =  ^Vt,  X  Th,  or  $.§ 0 ;   I  =  S8-70;   $8.70  ^  $^f^  = 

$8.70  X  ^r,  or  S580.,  Ans. 

EXAMPLE:     The  amount  due  at  the  maturity  of  a  30-day  6% 

note  is  $452.25;  what  is  the  principal? 
Interest  on  $1.00  for  30  days  at  6%  is  5  mills;  therefore,  each 
$1.00  amounts  to  $1,005.     If  all  the  principal  amounted  to 
$452.25  there  were  as  many  dollars  as  $1,005  is  contained  times 
in  $452.25. 

450  times;  therefore,  the  principal  is  $450.,  Ans. 
$1,005)  $452. 250 

A  -  (T  XR  4=  $1-00)  =  P 

T   X  R   =    360    X  JOTS)  or  $20  0;     *3  0  0    ~r  $2  0&    ~    $aoo; 

$452.25  -^    fU  =  $450.,  Ans. 


As  you  have  learned,  four  elements — principal,  rate 
per  cent,  time,  and  interest — are  concerned  in  every 
interest  example;  therefore,  if  any  three  of  these  ele- 
ments are  given,  the  fourth  can  be  found.  Thus,  in 
the  work  which  you  have  been  doing  you  have  always 
knov/n  the  principal,  the  rate  per  cent,  and  the  time, 
and  you  have  found  the  interest ^     P  X  R  X  T  =  I. 

To  find  the  principal  which  will  yield  a»  certain  amount 
of  interest  at  a  certain  rate  per  cent  in  a  given  time,  we 
divide  the  given  interest  by  the  interest  on  $1.00  for 

(vn-113) 


ARITHMETIC 

the  given  time  at  the  given  rate  per  cent,  and  as  the 
interest  on  $1.00  for  the  given  time  at  the  given  rate 
equals  T  X  R,  therefore,  I  -^  (T  X  R)  =  P. 

Exercise  64 — Oral. 

1.  The  interest  for  60  days  at  6%  is  what  per  cent 

of  the  principal? 

2.  If  we  know  the  interest  for  60  days  at  6%,  how 

can  we  find  the  principal?     Show  this  by  letters 
and  signs. 

3.  If  we  know  the  interest  for  90  days  at  6%,  how 

can  we  find  the  principal?     Show  this  by  letters 
and  signs. 

4.  If  we  know  the  interest  for  60  days  at  5%,  how 

can  we  find  the  principal?     Show  this  by  letters 
and  signs. 

5.  If  we  know  the  interest  for  120  days  at  5%,  how 

can  we  find  the  principal?    Show  this  by  letters 
and  signs. 

6.  What  per  cent  of  the  principal  is  the  amount  due 

at  the  maturity  of  a  60-day  6%  note? 

7.  If  the  correct  answer  to  Question  6  is  101%, 

and  you  know  the  amount  due  at  maturity,  how 
do  you  find  the  principal  in  such  an  example? 

8.  How  do  you  find  the  principal  of  a  30-day  5% 

note,  if  the  amount  due  at  maturity  is  known? 
Show  this  by  letters  and  signs. 

9.  What  four  elements  are  involved  in  every  interest 

example? 
10.  How  many  of  these  elements  must  be  known,  and 
how  many  may  be  unknown  in  any  example? 

(VII-114) 


PERCENTAGE 

Exercise  65 — Written. 

Solve,  and  prove  by  finding  the  interest : 
(Total  interest  -r-  interest  on  $1.  =  number  of  dollars 
in  principal;    I  -^  (T  X  R)  =  P). 
What  principal  will  yield: 

1.  $12.00  in  180  days  at  6%? 

2.  $2.25  in  30  days  at  5%? 

3.  $40.60  in  1  yr.  2  mo.  at  4%? 

4.  $156.25  in  2  yr.  6  mo.  at  5%? 

5.  $98.00  in  245  days  at  6%? 

Solve,  and  prove  by  finding  the  amount : 
(Total  amount  -^  amount  of  $1.  =  number  of  dollars 
in   principal;   A  -^  (T  X  R  +  $1.00)  =  P). 
WTiat  principal  will  amount  to: 

6.  $406.00  in  4  months  at  4|%? 

7.  $3,655.00  in  1  yr.  3  mo.  at  6%? 

8.  $429.27  in  180  days  at  5%? 

Lesson  30 
Finding  the  Time  When  the  Principal,  Rate,  and 


Interest  Are  Given 

EXAMPLE:     In  what  time  will  $420.00  yield  $1.75  at  5%? 
The  interest  for  1  year  at  5%  on  $420.00  =  $21.00; 
$1.75  -^  $21.00  =  ^1-1- (Ft),  or  ^  year  or  1  month,  Ans. 

$11  -f 

$21. 

7        11 
=  —  X  r~-,or  -7.  year  or  1  month,  Ans. 
4        aX         1^ 

P  X  R  = 

$420. 

3 

I  -^  (P  X  R)  =  T 
X  tItj.  or  $21.;  I  =  $1.75;   $1.75  h-  $21.  = 
3^  year  or  1  month,  Ans. 

(VII-llo) 


ARITHMETIC 

To  find  the  time  in  which  a  certain  principal  will  yield 
a  certain  amount  of  interest  at  a  certain  rate  per  cent, 
we  divide  the  total  interest  by  1  year's  interest  on  the 
given  principal  at  the  given  rate  per  cent,  and  as  1 
year's  interest  on  the  given  principal  at  the  given  rate 
per  cent  equals  P  X  R,  therefore,  I  -^  (P  X  R)  =  T. 


EXAMPLE:    In  what  time  will  $500.00  amount  to  S525.00  at  6%? 

Amount    =  $525.00    Interest  on  $500.00  for  1  yr.  at  6%  =  $30.00; 
Principal  =    500.00 

Sterest  "        ^^-^^      ^^^-^^  ^  ^^^-^^  ^  i  y^^^  ^^  ^^  ^^■'  ^"^• 


When  the  amount  due  at  maturity  is  given,  subtract 
the  principal  from  the  amount  to  find  the  total  interest, 
then  continue  as  before.     A  —  P  =  I . 

Exercise  66 — Written. 

Solve,  and  prove  by  finding  the  interest: 
(Total  interest  -^  interest  for  1  year  =  number  of 
years  in  time ;  I  -j-  (P  X  R)  =  T) . 
In  what  time  will: 

1.  $880.00  yield  $10.45  at  4f%? 

2.  $448.00  yield  $36.96  at  5i%? 

3.  $960.00  yield  $45.60  at  6^%? 

4.  $810.00  yield  $31.50  at  5%? 

5.  $416.50  yield  $83.30  at  6%? 

Solve,  and  prove  by  finding  the  amount : 
In  what  time  will : 

6.  $72.00  amount  to  $86.82  at  6^%? 

7.  $464.00  amount  to  $465.74  at  3%? 

(VII-116) 


PERCENTAGE 

8.  $876.00  amount  to  $883.30  at  5%? 

9.  $84.00  amount  to  $89.67  at  4|%? 

10.  $1,080.00  amount  to  $1,143.00  at  7%? 

Lesson  31 

Finding  the  Rate  Per  Cent  When  the  Principal, 
Time,  and  Interest  Are  Given 


EXAMPLE:     At  what  rate  per  cent  will  $450.00  yield  $11.25 
in  6  mo.? 

The  interest  on  $450.00  at  1%  for  6  months  =  $2.25;  therefore, 

the  ratemustbeas  great  as  $11.25  -r-  $2.25  =  ^W-,  or  5%,  Ans. 

I  -  (P  X  T)  =  R 

P  X  T  =  $450.  X  h  or  $225.;  I  =  $11.25;  $11.25  ^  $225.  = 

.05  or  5%,  Ans. 


To  find  the  rate  per  cent  at  which  a  certain  principal 
will  yield  a  certain  interest  in  a  certain  time,  we  divide 
the  total  interest  by  the  interest  at  1%  on  the  given 
principal  for  the  given  time,  and  as  the  interest  at  1%> 
on  the  given  principal  for  the  given  time  is  equal  to 
rio  of  (P  X  T),  therefore,  if  we  divide  I  by  (P  X  T) 
the  answer  will  be  in  hundredths  instead  of  per  cents. 


EXAMPLE:     At  what  rate  per  cent  will  $816.00  amount  to 

$907.80  in  2^  years? 
Amount  =  $907.80     The  interest  on  $816.00  at  1%  for  2^ 

Principal  =    816.00         years  equals  $20.40; 

Total  Interest  =  ~~9l^    $91.80  -i-  $20.40  =  H|§,  or4i%,  Ans. 


When  the  amount  due  at  maturity  is  given,  subtract 
the  principal  from  the  amount  to  find  the  total  interest, 
and  continue  as  before.     A  —  P  =  I . 

(VII-117) 


ARITHMETIC 

Exercise  67 — Oral. 

1.  If  we  know  that  1  year's  interest  on  a  certain 

principal  at  a  certain  rate  per  cent  is  $12.00,  in 
what  time  would  the  same  principal  yield  $6.00 
at  the  same  rate  per  cent? 

2.  In  what   time   would   the  same  principal  yield 

$24.00? 

3.  What  process  did  you  use  to  find  your  answers  to 

Question  1  and  Question  2? 

4.  In  what  time  will  $100.00  yield  $1.00  at  4%? 

5.  How  do  we  find  the  time  in  which  a  given  principal 

will  yield  a  certain  interest  at  a  certain  rate 
per  cent?     Show  this  by  letters  and  signs. 

6.  If  we  know  that  the  interest  at  1%  on  a  certain 

principal  for  a  certain  time  is  $8.00,  at  what 
rate  per  cent  would  the  same  principal  yield 
$40.00  in  the  same  tune? 

7.  What  process  did  you  use  to  find  your  answer  to 

Question  6? 

8.  At  what  rate  per  cent  will  $200.00  yield  $6.00  in 

1  year? 

9.  How  do  we  find  the  rate  per  cent  at  which  a 

certain  principal  will  yield  a  certain  interest  in 
a  certain  time?     Show  this  by  letters  and  signs. 

10.  When  finding  the  time  or  the  rate  per  cent,  what 

must  be  done  when  the  amount  due  at  maturity 
is  known,  but  the  interest  is  unknown? 

11.  Looking  at  Exercise  68,  state  how  you  will  prove 

your  answers  for  Examples  #1  to  #5. 

12.  Looking  at  Exercise  68,  state  how  you  will  prove 

your  answers  for  Examples  #6  to  #10. 
(Vii-ns) 


PERCENTAGE 

13.  Looking  at  Exercise  68,  state  how  you  will  prove 

your  answer  for  Example  #11. 

14.  Looking  at  Exercise  68,  state  how  you  will  prove 

your  answer  for  Example  #12. 

15.  Looking  at  Exercise  68,  state  how  you  will  prove 

your  answer  for  Example  #13.     State  how  you 
will  prove  your  answer  for  Example  #14. 

Exercise  68 — Written. 
Solve  and  prove: 

(Total  interest  -^  interest  at  1%  =  number  of  %  in 
rate;  I  -^  (P  X  T)  =  R). 

At  what  rate  per  cent  will : 

1.  $675.00  yield  $6.75  in  3  months? 

2.  $420.00  yield  $7.70  in  120  days? 

3.  $1,260.00  yield  $170.10  in  2  years  3  months? 

4.  $808.00  yield  $8.08  in  45  days? 

5.  $546.00  yield  $11.83  in  6|  months? 

At  what  rate  per  cent  will: 

6.  $4,800.00  amount  to  $4,809.00  in  15  days? 

7.  $72.00  amount  to  $74.10  in  210  days? 

8.  $4,124.00  amount  to  $4,216.79  in  9  months? 

9.  $42.00  amount  to  $42.98  in  4  mo.  20  da.? 
10.  $98.80  amount  to  $106.21  in  1  yr.  8  mo.? 

Are  you  sure  this  is  true  each  time : 

r  interest  on  $1  for  fuU  time,  at  full 

I      rate  =  principal. 

rp  .   ,  .    ,        +    .   J  interest  for  1  yr.  on  full  principal  at 
J-  ot/ai  interest  ~^  ^      <•  n  i  • 

full  rate  =  time. 

interest  at  1%  on  full  principal  for 

full  time  =  rate. 

(VII-119) 


ARITHMETIC 

11.  At  what  rate  will  $650.  yield  $97.50  interest  in  2 

yr.  6  mo.? 

12.  In  what  time  will  $922.50  yield  $49.20  interest 

at  4%? 

13.  How  much  money  loaned  for  3  yr.  8  mo.  at  5% 

will  yield  $112.20  interest? 

14.  In  what  time  will  $5,000.  yield  $5,000.  interest 

at  4%? 

Lesson  32 
Transposition  in  Figuring  Interest 


EXAMPLE:     Find  the  interest  on  $90.00  for  42  days  at  5%. 

Int. 
Int. 
Int. 
Int. 

on  $90.00  for  42  days  =  Int.  on 
for  60  days  at  6%  =  $0.42 
for  30  days  at  6%  =    0.21 
for  90  daj^s  at  6%  =  $0.63;  Int 
f  of  $0.63  =  $0.53, 

$42.00  for  90  days; 

.  at  5%  =  f  of  $0.63; 
Ans. 

You  will  remember  that  in  figuring  wages  we  can 

transpose  the  hours  worked  and  the  rate  per  week  to 

shorten    the    work,    when    the    rate  per  week  is  an 

aliquot  part  of  the  weekly  hour  basis;    as,  371  hours 

work  at  $16.00  per  week  on  a  48-hour  basis  is  the 

same  as  16  hours  at  $37.25  per  week,  the  answer  being 

$12.42  (i  of  $37.25). 

37^ 
if  of  $37.25  is  equal  to  -- -^  of  $16.00. 

48 

In  the  same  manner,  we  can  save  much  work  in 
figuring  interest  by  transposing  the  principal  and  the 
time  when  the  principal  is  an  aliquot  part  of  360, 
since  360  days  is  the  basis  on  which  commercial  interest 

(VII-120) 


PERCENTAGE 

is  figured;  thus,  in  figuring  the  interest  on  $60.00  for  197 
days  at  6%,  we  can  call  the  dollars  ^^days"  and  the  days 
''dollars,"  and  find  the  interest  on  $197.00  for  60  days 
at  6%,  the  answer  being  $1.97. 

^  X  yfc  X  $197.  is  equal  to  ifj  X  tJo  X  $60. 

Exercise  69 — Oral. 

1.  The  interest  on  $30.00  for  78  days  is  the  same  as 

the  interest  on  $78.00  for  how  many  days? 

2.  The  interest  on  $90.00  for  56  days  is  the  same  as 

the  interest  for  90  days  on  how  many  dollars? 

3.  How  can  you  find  the  interest  on  $60.00  for  273 

days  most  quickly? 

4.  What  is  the  interest  on  $60.00  for  273  days  at  6%? 

5.  When  can  work  be  saved  by  transposing  the  time 

and  the  principal  in  figuring  interest? 

6.  What  must  be  done  when  the  rate  is  more  or  less 

than  6%? 

7.  How  would  you  find  the  interest  on  $120.00  for 

18  days  at  5%?     See  if  you  can  give  the  answer. 

8.  How  would  you  find  the  interest  on  $45.00  for 

160  days  at  7%?   See  if  you  can  give  the  answer. 

9.  How  would  you  find  the  interest  on  $180.00  for 

411  days  at  6%? 
10.  How  would  you  find  the  interest  on  $90.00  for 
544  days  at  4^%? 

Exercise  70 — Written. 

Using  transposition,  find  the  interest  on: 

1.  $75.00  for  148  days  at  6%. 

2.  $15.00  for  312  days  at  8%. 

(VII-121) 


ARITHMETIC 

3.  $120.00  for  188  days  at  4^%. 

4.  $150.00  for  78  days  at  4%. 

5.  $80.00  for  96  days  at  5%. 

6.  $40.00  for  318  days  at  7%. 

7.  $180.00  for  1  day  at  6%. 

8.  $50.00  for  84  days  at  5^%. 

9.  $30.00  for  488  days  at  6^%. 
10.  $210.00  for  22  days  at  6%. 

Lesson  33 
Compound  Interest 


EXAMPLE :  Find  the  compound  interest  on  $500.00  for  3  years, 
at  6%,  the  interest  being  compounded  annually. 
$500.00  at  6%  =  $30.00  Interest  for  1st  year;  Amount  $530.00; 
$530.00  at  6%  =  $31.80  Interest  for  2d  year;  Amount  $561.80; 
$561.80  at  6%  =  $33.71  Interest  for  3d  year;  Amount  $595.51; 

Total $95.51  Compound  Interest  for  3  years. 

or: 

Final  Amount $595.51 

Original  Principal 500.00 

$95.51  Compound  Interest  for  3  years. 


When  the  interest  for  stated  periods  is  added  to  the 
principal  and  the  amount  so  found  is  used  as  the 
principal  for  the  next  interest  period,  the  total  interest 
so  added  to  the  several  principals  is  called  '^compound 
interest." 

Savings  banks  usually  allow  compound  interest, 
adding  the  interest  to  the  principal  quarterly  or  semi- 
annually to  form  each  new  principal.  Try  to  find  out 
how  the  savings  banks  in  your  locality  pay  interest. 

(VII-122) 


PERCENTAGE 

Bear  in  mind  that : 

6%  annually  =  3%  semi-annually,  or  1^%  quarterly. 
5%  annually  =  2 1%  semi-annually,  or  lj%  quarterly. 
4%  annually  =  2%  semi-annually,  or  1%  quarterly. 
3%  annually  =  1  §%  semi-annually,  or    f%  quarterly. 

Exercise  71 — Oral. 

1.  In  figuring  compound  interest,  how  often  does  the 

principal  change  in  a  year  if  the  interest  is  pay- 
able annually? 

2.  In  figuring  compound  interest,  how  often  does  the 

principal  change  in  a  year  if  the  interest  is  pay- 
able semi-annually? 

3.  In  figuring  compound  interest,  how  often  does  the 

principal  change  in  a  year  if  the  interest  is  pay- 
able quarterly? 

4.  What  is  the  longest  period  of  time  that  one  prin- 

cipal  can  be  used  if  the   interest   is  payable 
quarterly? 

5.  What   is   the   longest   period   of   time   that   one 

principal  can  be  used  if  the  interest  is  payable 
semi-annually? 

6.  What  is  the  longest  period  of  time  that  one  prin- 

cipal  can   be  used   if   the  interest   is  payable 
annually? 

7.  How  do  savings  banks  usually  pay  interest? 

8.  What  is  the  difference  between  the  final  amount 

and  the  original  principal  called? 

9.  In  figuring  the  compound  int'erest  on  $375.00  for 

18  months  at  4%   compounded  semi-annually, 
how  many  separate  interest  calculations  must 

(111-123) 


ARITHMETIC 

yc'U  make?     How  long  a  period  of  time  will  be 
covered  by  each  interest  calculation? 
10.  State  how  you  would  find  the  compound  interest 
on  $1,000.00  for  2  years  at  3%  payable  quarterly. 

Exercise  72 — Written. 

Find  the  compound  interest  on : 

1.  $450.00  for  3  years  at  5%  payable  annually. 

2.  $600.00  for  2  years  at  4%  payable  semi-annually. 

3.  $1,000.00  for  9  months  at  3%  payable  quarterly. 

4.  $275.00  for  1  year  at  4%  compounded  quarterly. 

5.  What  will  $75.00  amount  to  in  6  months  at  3% 

compounded  quarterly? 


(V 11-124) 


<i 


ACCOUNTS 

Lesson  34 

Savings  Bank  Accounts 

When  money  is  placed  in  a  bank,  it  is  said  to  be 

deposited'';  when  it  is  taken  out  of  a  bank,  it  Is  said 
to  be  ''withdrawn";  the  amount  remaining  on  deposit 
at  any  time  is  called  the  ''balance." 

When  a  "savings  account"  is  opened,  the  bank  gives 
the  depositor  a  bank  book  in  which  an  entry  is  made 
to  show  the  amount  deposited;  thereafter  when  money 
is  deposited  or  withdrawTi,  this  bank  book  must  be 
presented  so  that  the  amount  deposited  or  withdrawn 
can  be  recorded  and  the  new  balance  shown. 

Savings  banks  usually  pay  interest  at  3%,  3|%  or 
4%,  compounded  quarterly  or  semi-annually,  but  they 
deduct  amounts  withdrawn  during  any  interest  period 
from  the  balance  at  the  beginning  of  the  period  if 
possible,  otherwise  from  the  first  deposits  made;  thus, 
an  amount  must  be  left  on  deposit  to  the  end  of  the 
interest  period  to  earn  any  interest. 

All  deposits  made  during  the  first  few  business  days 
of  any  month  draw  interest  from  the  first  of  that 
month;  other  deposits  draw  interest  from  the  first  of 
the  following  month,  and  no  interest  is  paid  on  frac- 
tions of  a  dollar. 

In  the  savings  account  here  shown,  the  withdrawals, 
amounting  to  $185.00,  would  be  applied  against  the 
deposits  of.  March  3,  $150.00,  and  March  31,  $40.00, 

(VII-125) 


ARITHMETIC 


leaving  only  $5.00  on  which  interest  would  be  paid 
from  April  1st,  and  $43.00  on  which  interest  would  be 
paid  from  May  1st  to  July  1st  which  is  the  end  of  the 
semi-annual  period,  and  this  at  3%  amounts  to  $0.24 
which  is  entered  as  a  deposit  and  added  to  the  balance. 


r  ^ 

Dr.  NATIONAL  TRUST  &  SAVINGS  BANK 

CHICAGO,  ILLINOIS 

In  account  ^[^h:*  x^t\        No.  J/S'j  ^^<^ 

'<^€y     Ly z^^^C^^^tL^^^^  Cr. 


Date 


/9^ 


Initial 


Withdrtwak 


^r^o 


Zao 


\Jh^i^  ^^c^^^^^ 


0  6 


Deposits 


/^^ 


^0 


^S 


0  0 


00 


^ 


Balance 


/  ^^ 


Gq 


/ 0^  00 


/y^ 


^^ 


^_^ 


00 


^6 


~0 


s~o 


7jI 


A  Page  from  a  Savings  Bank  Book. 

Exercise  73 — Oral. 

1.  On    the   bank   book   shown   in   the   illustration, 

which  entries  refer  to  amounts  put  into  the  bank? 

2.  Which  entries  refer  to  amounts  taken  out  of  the 

bank? 

(VII-126) 


ACCOUNTS 

3.  What  is  meant  by  a  '^ deposit"? 

4.  What  is  meant  by  a  ^'withdrawal"? 

5.  What  is  meant  by  a  "balance"? 

6.  At  what  rate  per  cent  do  savings  banks  usually 

pay  interest?  What  kind  of  interest  do  they 
pay? 

7.  If  an  amount  is  withdrawn  before  the  end  of  the 

interest  period,  is  any  interest  paid  thereon  for 
the  time  it  was  on  deposit? 

8.  When  does  an  amount  deposited  during  the  first 

few  da^^s  of  a  month  start  to  draw  interest? 
When  does  an  amount  deposited  at  any  other 
time  start  to  draw  interest? 

9.  What  name  is  given  any  sum  of  money  put  into 

a  bank? 

10.  What  name  is  given  any  sum  of  money  taken 

out  of  a  bank? 

11.  What  name  is  given  to  the  amount  you  have  re- 

maining in  a  bank  at  any  time? 

12.  Why  is  a  bank  book  given  by  the  bank  to  the 

depositor  when  a  savings  account  is  opened? 

13.  Why  must  the  bank  book  be  presented  every 

time  money  is  deposited  or  withdrawn? 

14.  Explain   what   is  meant  when  we   say   that   a 

savings  bank  pays  interest  quarterly? 

15.  Explain   what   is   meant   when   we    say   that   a 

savings  bank  pays  interest  semi-annually? 

16.  Is  any  interest  paid  on  fractions  of  a  dollar? 

17.  How  is  the  interest  paid  to  the  depositor? 

18.  If  you  have  $100.  on  deposit  in  a  savings  bank 

which  pays  interest   quarterly  at  the  rate  of 
C\^II-127) 


ARITHMETIC 

4%  per  annum,  how  much  interest  could  you 
withdraw  every  three  months?  Would  the 
withdrawing  of  this  interest  every  three  months 
reduce  your  $100.  balance?     Explain  fully. 

Exercise  74 — Written. 

1.  Rule  paper  to  represent  a  savings  bank  book, 

record  the  following  transactions,  and  find  the 
balance  of  the  account : 
Deposit  April  6,  $150.00; 
Deposit  May  14,  $100.00; 
Withdrawal  June  9,  $75.00. 

2.  If  the  bank  in  Question  1  pays  interest  at  3% 

compounded  semi-annually  on  Jan.  1st  and  July 
1st,  and  all  deposits  made  on  or  before  the  5th 
of  any  month  draw  interest  from  the  first  of  that 
month,  what  amount  of  interest  will  be  credited 
on  this  account  July  1st?  Make  the  necessary 
entry  on  the  account. 

3.  Rule  paper,  record  the  following  transactions,  and 

find  the  balance  of  the  account : 

Deposits:  July  3,  $1,400.00;  Sep.  6,  $200.00; 

Withdrawals:  Aug.  1,  $300.00;  Nov.  1,  $400.00. 

4.  If  the  bank  in  Question  3  pays  interest  at  4% 

compounded  semi-annually  Jan.  1st  and  July 
1st,  and  all  deposits  made  on  or  before  the  5th  of 
any  month  draw  interest  from  the  first  of  that 
month,  what  amount  of  interest  will  be  credited 
on  this  account  Jan.  1st?  Make  the  necessary 
entry  on  the  account. 

(VII-128) 


ACCOUNTS 

Lesson  35 
Bank  Accounts  Which  Are  Subject  to  Check 

Bank  accounts  on  which  checks  can  be  issued,  or 
"checking  accounts,"  as  they  are  very  often  called, 
are  used  by  individuals,  firms,  and  companies  to  conduct 
their  finances. 

Money  deposited  in  these  accounts  is  entered  in  the 
depositor's  bank  book,  but  draws  no  interest  (excepting 
in  special  cases). 

To  Tvdthdraw  money,  the  depositor  is  not  required  to 
present  his  bank  book  as  in  the  case  of  savings  accounts : 
instead,  he  issues  checks  for  the  amounts  he  wishes  to 
withdraw  and  makes  them  payable  to  the  parties  to 
whom  he  wishes  to  pay  the  money. 

These  checks,  after  being  cashed  by  the  bank,  are 
charged  to  the  depositor's  account  and  are  returned  to 
him  with  a  statement  of  his  account  on  the  first  day 
of  each  month. 


No.  /^/   \    No.  /y/  Chicago,    O^^^^-    ^ 19.2^ 

Date    (^y3^/^o I 

Balance       ^^M  FIRST  NATIONAL  BANK 

Deposit  <^/^^     ^a.oA  OF  CraCAGO.  ILLINOIS 

Total ^C^od 

Check  payable 

Amount d'coo 

Balance  ^/  :f:o(^  \ 


Pay  to  the  order  «/  /'KcX{j'^^^^j(--^^-€/c^^-l^_$. 


o  o^ 


'-'f  -y^a Dollars 


The  ''payee"  of  a  check  is  the  party  to  whom  it  is 

payable. 

CYn-129) 


ARITHMETIC 


Reconcilement  of  a  Bank  Account 

June  30,  1920 

Balance  as  per  Check  Book, 

$415.00 

Checks  Outstanding: 

#140     $75.00 

#  141       50.00 

125.00 

Balance  as  per  Bank 

Statement 

$540.00 

Since  the  bank  does  not  know  of  the  existence  of  any 
check  until  it  is  presented  for  payment,  any  checks  not 
cashed  on  the  last  day  of  a  month  would  naturally  not 
be  charged  by  the  bank  against  the  depositor's  account; 
therefore,  to  make  his  account  agree  with  the  bank,  the 
depositor  must  add  all  uncashed  outstanding  checks  to 
his  balance.  Balancing  in  this  manner  is  called  a 
^^reconcilement"  of  a  bank  account. 

To  ascertain  what  checks  are  outstanding  uncashed 
at  the  end  of  the  month,  all  cashed  checks  received  from 
the  bank  must  be  sorted  numerically  and  compared 
with  the  stubs. 

Exercise  75 — Oral. 

i.  In  the  check  here  illustrated,  who  is  paying  a  sum 
of  money?  Who  is  receiving  a  sum  of  money? 
What  sum  of  money  is  being  paid? 

2.  What  has  the  First  National  Bank  to  do  with  this 

transaction? 

3.  What  will  the  First  National  Bank  do  with  this 

check  after  paying  it? 

4.  Who  is  the  payee  of  this  check? 

(vn-130) 


ACCOUNTS 

5.  Supposing  you  had  a  bank  account  and  issued  a 

check  on  April  30th  for  SI 00. 00  payable  to  a 
party  in  another  city  and  mailed  it  to  him, 
would  your  bank  be  able  to  cash  this  check 
during  April?  When  would  they  be  likely  to 
cash  it? 

6.  In  the  case  mentioned  in  Question  5,  how  would 

you  be  able  to  reconcile  your  bank  account  on 
April  30th? 

7.  "\Miat  is  meant  by  reconciling  a  bank  account? 

8.  How  can  you  ascertain  which  checks  are  out- 

standing unpaid  at  the  end  of  the  month? 

9.  Explain  each  of  the  entries  on  the  stub  shown  in 

the  illustration. 
10.  Explain  each  of  the  entries  in  the  reconcilement 
shown  in  the  illustration. 

Exercise  76 — Written. 

1.  Find  the  balance  as  of  September  30th  on  the 

following  bank  account: 

Balance  Sep.    1 $1,412.75; 

Deposit  Sep.    4 473.86; 

Checks  Issued: 

#17,474  Sep.    4 87.13 

75  Sep.    8 146.87 

76  Sep.  15 468.72 

77  Sep.  29 321.62 

78  Sep.  30 45.00. 

2.  What  balance  would  the  bank  show  on  the  account 

given  in  Question  1   Sep.  30,  if  checks  #17,477 

(VII-131) 


ARITHMETIC 

and  #17,478  were  sent  to  distant  cities  and  there- 
fore were  not  cashed  until  early  in  October? 

3.  Prepare  a  reconcilement  of  the  bank  account  given 

in  Question  1  and  Question  2  as  of  Sep.  30th. 

4.  Find  the  balance  as  of  Dec.  31st  of  the  following 

bank  account: 

Balance  Dec.    1 $18,745.36; 

Deposits : 

Dec.    5 1,486.32 

Dec.  13 2,394.97 

Dec.  20 1,663.13 

Checks  Issued: 

#11,416  Dec.    3... 1,741.32 

17  Dec.    3 ;  431.87 

18  Dec.  19 296.98 

19  Dec.  28 1,347.21 

20  Dec.  28 841.38 

21  Dec.  30 711.48 

22  Dec.  30 222.47. 

5.  On   Dec.    31st  the  bank   returned  the  following 

checks  with  a  monthly  statement  of  the  account 
given  in  Question  4 : 

#11,416; 
17; 

18; 
20; 
What  balance  appeared  on  this  statement? 

6.  Construct  a  reconcilement  of  the  bank  account 

given  in  Question  4  and  Question  5  as  of  Dec. 
31st. 

(VII-132) 


ACCOUNTS 

7.  losing  the  form  shown  in  the  iUustration,  write  a 

check  dated  New  York,  Oct.  3,  1918,  drawn  on 
the  Citizens'  National  Bank  of  New  York  for 
$100.00,  payable  to  Harry  Knowles,  signed  by 
yourself. 

8.  AYrite  a  check  dated  today  showing  how  you  would 

pay  your  teacher  $15.46  if  you  had  money  on 
deposit  at  the  Continental  Bank  of  Cincinnati, 
Ohio. 

9.  Write  a  check  dated  today  showing  how  Wilson 

&  Co.  of  Philadelphia,  Pa.,  could  pay  you  $18.75 
if  they  had  money  on  deposit  at  the  Old  Trust 
Company  of  Boston,  Mass. 
10.  Write  a  check  on  the  First  Bank  and  Trust  Com- 
pany of  San  Francisco,  Calif.,  in  the  sum  of 
$1,000.00,  with  your  teacher  as  the  payee,  using 
today's  date. 

Exercise  77 — Oral  Review. 

1.  What  is  the  interest  on  $60.00  for  318  days  at  6%? 

2.  What  are  the  dimensions  of  a  township?     How 

many  sections  does  a  township  contain? 

3.  W^hat  is  the  ratio  of  the  circumference  of  a  circle 

to  the  diameter? 

4.  If  a  teamster's  w^ages  are  $24.00  for  a  week  of  48 

hours,  how  much  will  he  receive  for  working 
18"i  hours? 

5.  Multiply: 

(a)  (h)  (c) 

64X37 J;  56X.75;  66  X  33i 

(VII-133) 


ARITHMETIC 

6.  Divide: 

(a)  (h)  (c) 

600  -^  1()|;  11  -^  25;  60  -^  33i 

7.  What  is  the  commission  on  the  sale  of  produce 

amounting  to  $400.00  at  12^%? 

8.  What  is  the  circumference  of  a  circle  if  the  diam- 

eter is  10  inches? 

9.  What  is  the  perimeter  of  a  10-yard  square?    What 

is  the  area? 

10.  How  many  cubic  inches  are  there  in  a  prism  2  in. 

wide,  3  in.  thick,  and  12  in.  high? 

11.  What  is  the  volume  of  a  10-inch  cube?     What 

is  the  area  of  its  entil'e  surface? 

Exercise  78    Written  Review. 

1.  What  is  the  interest  on  $346.00  from  June  9,  1918, 

to  Aug.  14,  1920,  at  6%? 

2.  What  is  the  volume  of  a  section  of  stove  pipe  6" 

in  diameter  and  20  inches  long? 

3.  What  is  the  area  of  the  curved  surface  of  5  sections 

of  this  stove  pipe? 

4.  Find  the  net  amount  of  an  invoice  of  $178.00 

subject  to  10%  and  10%  discount. 

5.  The  discounts  on  an  invoice  are  equivalent  to 

27.1%;   if  two  of  the  rates  are  10%  and  10%, 
what  is  the  other  rate? 

6.  In  what  time  will  $87.60  amount  to  $88.33  at  5% 

interest? 

7.  At  what  rate  per  cent  will  $126.00  amount  to 

$143.01  in  2  yr.  3  mo.? 
(VII-134) 


ACCOUNTS 


8.  Find  the  compound  interest  on  $2,750.00  for  1  yr. 

at  4%  payable  quarterly. 
Add,  but  do  not  copy: 
(Time  for  these  3  examples  is  less  than  4§  minutes.) 


9. 

10. 

11. 

1,974 

4,612 

7,387 

4,132 

3,734 

6,219 

7,385 

5,875 

7,385 

3,198 

6,298 

5,873 

5,988 

7,129 

9,821 

3,085 

3,975 

6,185 

7,142 

7,309 

5,318 

1,251 

7,111 

3,071 

2,038 

5,873 

6,936 

6,712 

4,109 

3,174 

1,788 

1,291 

2,682 

9,174 

3,712 

8,638 

4,782 

1,267 

5,021 

Copy  and  multiply: 
(Time  for  these  8  examples  is  less  than  4|  minutes.) 


12.     1,812  X  35; 

16. 

7,315  X  63; 

13.    4,387  X  43; 

17. 

1,288  X  92; 

14.     3,815  X  28; 

18. 

5,167  X  84; 

15.     6,345  X  52; 

19. 

7,319  X  27. 

Copy  and  divide: 

(Time  for  these  6  examples 

is  less  than  4|  minutes.) 

20.     14,620  4-  34; 

23. 

35,776  -^  43; 

21.     19,328  -^  32; 

24. 

56,820  -7-  60; 

22.     39,273  -^  53; 

25. 

33,054  ~  42. 

(VII-135) 


ARITHMETIC 

Subtract,  but  do  not  copy: 
(Time  for  these  12  examples  is  less  than  4 J  minutes.) 
26.  27.  28.  29. 

353,872  475,318  598,712  843,829 

289,746  329,879  231,987  421,945 


30. 

31. 

32. 

33. 

611,841 

741,028 

681,491 

831,874 

298,748 

263,751 

298,728 

219,008 

34.   , 

35. 

36. 

37. 

439,821 

288,487 

912,091 

210,005 

318,941 

179,209 

138,099 

153,901 

(VII-136) 


ADVANCED  LESSONS 

PART  vni 

NOTATION   AND   NUMERATION 

Lesson  1 
The  Higher  Periods 

Write  eight;  eight  hundred;  eight  thousand;  eight 
miUion.     Which  of  these  is  the  largest?     Tell  why. 

Each  of  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  9  has  two  values; 
one  of  these  values  is  absolute  and  is  expressed  by  the 
form  of  the  figure;  the  other  value  is  relative  and 
depends  upon  the  position  that  the  figure  occupies  in 
a  number.  Thus,  the  figure  ^^8"  always  has  an  abso- 
lute value  of  ''eight,"  but  in  the  number  ''894"  the 
figure  "8,"  besides  having  an  absolute  value  of  "eight," 
also  has  a  relative  value  of  "eight  hundred,"  since  it 
occupies  hundreds'  place  in  this  particular  number. 

Read  evenly: 

80,452;  995,800: 

912,816;  1,265,104 

999,999;  9,999,999 

1,000,000;  10,000,000 

In  the  same  manner  as  1,000  follows  999  and  1,000,000 
follows  999,999,  so  we  can  have  any  number  of  places 
and  periods,  but  more  than  five  periods  are  seldom 
necessary  to  express  a  numerical  value,  as  few  things 
are  so  vast  in  their  scope  that  more  periods  are  neces- 
sary. 

(VIII-l) 


65,100,142 

99,999,999 

100,000,000 

999,999,999 . 


ARITHMETIC 

The  first  five  periods — with  three  of  which  you  are 
already  entirely  familiar — and  the  names  of  the  places 
they  comprise  are  as  follows: 


The  -S  -S  -S  -^     „, 

Places         Is^gg        Is-sSS        ISoSS       l^-^iS         -? 

or  -i-S^.sl         |§S.og         -|.2°|.2         IS^SS  -I     «    S 

^  C—    fi—  S  S'^C^*^  C3C=33  flOflOo  "^d-- 

Orders        jgg^gg        jgg^Ss        ^-^^^jl        ^^-^^  ^g    ^    g 

846,     398,     746,     872,     361 

Periods         Trillions        Billions         Millions      Thousands         Units 

The  names  of  the  next  seven  periods  in  their  order 
are:  quadrillions,  quintillions,  sextillions,  septillions, 
octillions,  nonillions,  decillions. 

Exercise  1 — Oral. 

1.  In  the  number  764,384  what  is  the  absolute  value 

of  each  of  the  digits? 

2.  In  the  number  764,384  what  is  the  relative  value 

of  each  4? 

3.  What  determines  the  absolute  value  of  a  digit? 

4.  What  determines  the  relative  value  of  a  digit? 

5.  What  is  the  name  of  the  first  place  of  the  3d  period 

(  6,000,000)?  What  order  is  this  when  units' 
place  is  counted  as  the  first?  What  places  make 
up  this  full  period? 

6.  What  is  the  first  place  of  the  4th  period  called? 

What  places  make  up  this  period? 

7.  What  is  the  first  place  of  the  5th  period  called? 

What  places  make  up  this  period? 

8.  What  is  the  first  place   in   any  period  called? 

The  second  place?     The  third  place? 
(VIII-2) 


NOTATION   AND   NUMERATION 

9.  Name  5  periods;  begin  with  units'  period. 
10.  Beginning   with   units,   name   all   the  places   to 
trillions'  place. 

Exercise  2 — Oral. 

Point  off,  and  read  the  following  statements  smoothly : 

1.  Rays  of  heat  from  the  sun  travel  94,500,000  miles 

before  they  reach  us  in  June;  they  travel 
91,500,000  miles  in  January. 

2.  The  distance  across  the  earth's  orbit  is  186,000,000 

miles  the  long  way. 

3.  The  area  of  the  earth's  surface  is  about  197,000,000 

sq.  mi. 

4.  The  earth  has  about  1,700,000,000  people. 

5.  During  the  World  War  it  was  necessary  for  the 

United  States  to  borrow  money  by  issuing 
Liberty  Bonds.  The  subscriptions  for  the 
Fourth  Liberty  Loan  exceeded  $6,000,000,000.00. 

6.  During  the  five  years  1912-1916  inclusive,  the 

average  yearly  production  of  the  United  States 
was: 

2,761,252,000  bu.  com. 

1,296,406,000  bu.  oats. 

7.  The  lumber  cut  in  the  United  States  during  the 

year  1917  was  35,821,239,000  ft. 

8.  To  June  29,  1918,  the  Senate  of  the  United  States 

passed  war  bills  aggregating  $21,500,000,000.00 
for  conducting  the  World  War. 

9.  The  national  wealth  of  the  United  States  including 

all   the   real   and   personal   property   of   every 

(VIII-3) 


ARITHMETIC 

description  was  estimated  to  be  $187,739,000,- 
000.00  in  the  year  1912. 
10.  Before  the  war  the  national  wealth  of  France  was 
approximately  $63,000,000,000.  and  the  national 
income  about  $7,500,000,000. 

Exercise  3 — Written. 

Write: 

1.  Eight  hundred  forty-six  billion,   three  hundred 

seventy-two    million,    one    hundred    forty-five 
thousand,  nine  hundred. 

2.  Four    hundred    sixteen    billion,    three    hundred 

twenty-five  thousand,  four  hundred  sixty-two. 

3.  Eight  hundred  seventy-four  trillion,  three  hundred 

twenty-one    billion,    four    hundred    sixty-two 
million,  one  hundred. 

4.  Six  hundred  five  trillion,  four  hundred  forty-five 

million,  eight  hundred  thirty-one. 

5.  Thirty-eight    trillion,    seven    hundred    forty-six 

billion,  one  hundred  forty-five. 

6.  Three  hundred  eighty-six  trillion,  four  hundred 

ninety-eight  million,  two  hundred  twenty  thous- 
and, one. 

7.  Three  hundred  seven  trillion,  thirty-seven  billion, 

thirty  million,  seven  hundred  thousand,  thirty. 

8.  Eight    hundred    twelve    billion,    eight    hundred 

million,  twelve  thousand,  eight  hundred  twelve. 

9.  Sixty  trillion,  six  hundred  billion,  six  million,  six 

hundred  thousand,  sixty-six. 
10.  Three  hundred  forty-five  trillion,  nine  hundred 
eleven  thousand. 

(VIII-4) 


DENOMINATE  NUMBERS 

Lesson  2 

Table  of  Circular  Measure 

60  seconds  {")  =  1  minute  (') 
60  minutes. . .  =^  1  degree  (°) 
90  degrees.  .  .  =  1  quadrant  (quad.) 
360  degrees.  .  .  =  1  circle  (O) 


A  Quadrant 


or 


Quadrant 

or 
I  Circle 


Since  there  are  360  degrees  in  every  circumference, 
a  degree  of  arc  is  always  ^g-o  of  a  complete  circumfer- 
ence, regardless  of  the  size  of  the  circle.  As  a  circum- 
ference may  be  an  inch,  a  foot,  a  yard,  a  mile,  or  25,000 
miles  in  length,  one  degree  of  arc  will  represent  as 
many  different  distances  as  there  are  different  cir- 
cumferences, because  the  number  of  inches,  yards,  etc., 
in  a  degree  varies  with  the  size  of  the  circle,  but  the 
number  of  degrees  in  every  circumference  always  is  360 
regardless  of  the  size  of  the  circle.  Remember  that  a 
circumference  1  inch  in  length  contains  just  as  many 
degrees  as  one  25,000  miles  in  length. 

(vin-5) 


ARITHMETIC 

For  making  the  more  accurate  measurements  in  sur- 
veying and  astronomy  each  degree  is  divided  into  60 
equal  parts  called  minutes,  and  each  minute  is  divided 
into  60  equal  parts  called  seconds. 

Exercise  4 — Oral. 

1.  Draw  a  circle.     How  many  degrees  are  there  in 

your  circle? 

2.  If  each  one  in  the  class  draws  a  circle,  does  each 

circle  contain  360°? 

3.  Does  the  size  of  the  circle  make  any  difference? 

4.  How  many  degrees  are  there  in  J  of  a  circle?  In 

1  of  i?      _ 

5.  One  degree  is  what  part  of  a  circumference? 

6.  If  the  circumference  is  25,000  miles,  1°  of  that  is 

found  how? 

7.  If  the  circumference  is  9  inches,  how  do  you  find 

the  length  of  1°? 
The  ''horizon"  is  the  line  where  the  sky  and  earth 
seem  to  meet. 

8.  How  many  degrees  of  arc  are  there  in  the  entire 

horizon? 

9.  How  many  degrees  of  arc  are  there  in  the  horizon 

between  points  exactly  east  and  exactly  west? 
How  many  degrees  between  points  exactly 
north  and  exactly  south? 

10.  Point  with  your  arm  toward  the  eastern  horizon. 

What  is  the  position  of  your  arm  while  pointing? 
The  point  directly  overhead  is  called  the  '^zenith." 

11.  How  many  degrees  of  arc  are  there  between  any 

point  on  the  horizon  and  the  zenith? 
(VIII-6) 


DENOMINATE   NUMBERS 

12.  From  horizon  to  horizon  through  the  zenith  is 

how  many  degrees?  Does  the  direction  of  such 
an  arc  make  any  difference? 

13.  Point  with  your  arm  toward  the  zenith.     "WTiat 

is  the  position  of  your  arm  while  it  is  pointing 
toward  the  zenith? 

14.  How  many  degrees  are  there  between  the  hands 

of  the  clock  when  it  is  3  o^clock? 

15.  If  a  star  is  30°  from  the  zenith  on  an  arc  toward 

the  southern  horizon,  how  many  degrees  is  it 
from  the  southern  horizon?  Point  with  your 
arm  in  the  direction  where  such  a  star  would  be. 
What  is  the  position  of  your  arm  while  it  is 
pointing  toward  this  star? 

Exercise  5 — Written. 

1.  How  many  degrees  of  arc  are  there  in  J  circum- 

ference? How  many  minutes  of  arc?  How 
many  seconds  of  arc? 

2.  Reduce  35,745"  to  °  '  ". 

3.  f  \  Circumference  =  28,800  miles;  1°  =  ? 


4.  A^ jB         j  Find  the  length  of  x. 


5,  Change  2°  15'  15"  to  seconds. 

(VIII-7) 


ARITHMETIC 

Lesson  3 
Longitude  and  Time 


WESTERN  HEMISPHERE  EASTERN  HEMISPHERE 

The  World 

Besides  measuring  distances  on  the  earth^s  surface 
in  units  of  miles,  rods,  etc.,  we  can,  since  the  earth  is 
almost  a  sphere,  measure  these  distances  along  arcs  of 
imaginary  circles,  and  express  them  in  units  of  degrees, 
minutes,  and  seconds  of  arc,  in  the  same  manner  as  we 
measure  other  circles. 

The  equator,  being  a  circumference  of  the  earth, 
contains  360°  and  each  of  these  degrees  is  1°  of  longi- 
tude. The  0°  point  on  the  equator  is  on  an  arc  (called 
the  Prime  Meridian)  drawn  from  pole  to  pole  through 
Greenwich,  near  London,  England,  where  one  of  the 
world's  greatest  observatories  is  located,  and  all  places 
are  East  or  West,  depending  upon  their  location  on 
other  '^meridians  of  longitude"  east  or  west  of  this 
Prime  Meridian.     No  point  can  be  called  more  than 

(VIII-8) 


DENOMINATE   NUMBERS 

180®  east  or  west  of  Greenwich,  since,  if  it  were  190"^ 
east  it  would  be  only  170°  west  and  it  would  be  so 
called. 

There  are  other  imaginary  lines  running  parallel  to 
the  equator  to  show  distances  north  and  south  of  the 
equator,  and  these  are  called  '^parallels  of  latitude." 

All  places  on  the  same  meridian  have  the  same 
longitude;  all  places  on  the  same  parallel  have  the 
same  latitude. 

Exercise  6 — Oral. 

Follow  Harry's  imaginary  trips  and  troubles. 

1.  He  is  traveling  on  an  imaginary  circle  equally 

distant  from  the  poles;  what  imaginary  circle  is 
this? 

2.  He  has  a  trip  of  how  many  degrees  if  he  tries  to 

reach  either  of  the  poles? 

3.  Longitude  is  distance  east  or  west  of  the  Prime 

Meridian.  If  he  is  where  the  Prime  Meridian 
crosses  the  equator,  write  the  longitude  of  his 
place. 

4.  If  he  travels  the  circumference  of  the  equator 

how  many  degrees  does  he  travel? 

5.  How  many  °  can  he  travel  to  the  greatest  east 

longitude?  How  many  to  the  greatest  west 
longitude? 

6.  Why  is  it  wrong  to  say  260°  East? 

7.  He  visits  several  cities  that  have  east  longitude. 

Name  3. 

8.  Follow  him  to  3  cities  in  west  longitude.     Name 

them. 

(VIII-9) 


ARITHMETIC 

9.  Follow  him  to  the  cities  that  have  approximately 
the  following  locations,  and  tell  what  cities 
they  are: 

(a)  90°  West  Longitude;  30°  North  Latitude. 
(6)   105°  West  Longitude;  40°  North  Latitude. 

(c)  30°  East  Longitude;  30°  North  Latitude. 

(d)  0°  East  Longitude;  50°  North  Latitude. 

10.  From  the  map  tell  his  longitude  if  he  visits  Buenos 
Aires. 

Exercise  7 — Oral. 

Harry  studies  the  map  and  clock.  Since  the  earth 
rotates  on  its  axis  from  west  to  east  once  every  24 
hr.,  he  has  some  questions  to  answer.  Can  you  help 
him? 

1.  (a)     24  hr.  of  time  corresponds  to  360°  longitude; 

(b)  1  hr.  of  time  corresponds  to    ?    longitude; 

(t^  of  360°) ; 

(c)  1  min.  of  time  corresponds  to    ?    longitude ; 

(A  of  15°); 

(d)  1  sec.   of  time  corresponds  to    ?    longitude; 

(^  of  15'). 

2.  (a)  360°  longitude  corresponds  to  24  hr.  of  time; 
(6)       1°  longitude  corresponds  to  ?  min.  of  time; 

(sio  of  24  hr.) ; 

(c)  1' longitude  corresponds  to  ?  sec.    of  time; 

{wu  of  4  min.) ; 

(d)  1"  longitude  corresponds  to  ?  sec.   of  time; 

(^  of  4  sec). 

3.  (a)     15°  longitude  corresponds  to  how  much  time? 
(h)     15'  longitude  corresponds  to  how  much  time? 

(VIII-IO) 


DENOMINATE   NUMBERS 

(c)  15"  longitude  corresponds  to  how  much  time? 

(d)  15°   15'   15"  longitude  corresponds  to  how 

much  time? 

4.  How  many  of  each  unit  of  longitude  correspond 

to  one  of  each  unit  of  time?     1  hr.,  1  min.,  1 

sec.  =  r,  r,  r. 

5.  If  there  is  a  difference  in  time  between  two  places, 

what  process  will  find  the  corresponding  differ- 
ence in  longitude? 

6.  As  New  York  is  about  15°  east  of  Chicago,  which 

city  sees  the  sun  first?    How  many  hours  sooner? 

7.  How  far  are  two  cities  apart  if  A  is  30°  East  and 

Bis  60°  West?  Did  you  add  or  subtract?  Why? 
What  is  the  difference  in  time? 

8.  If  M  is  90°  E.  and  N  is  105°  E.,  what  is  the  differ- 

ence in  longitude?     Add  or  subtract?     WTiy? 

9.  How  many  degrees  of  longitude  correspond  to  3 

hr.  of  time?     2\  hr.? 

10.  Two  cities  are  90°  apart;   what  is  the  difference 

in  time? 

11.  A  has  80°  12'  W.  longitude;  B  has  10°  3'  E.  longi- 

tude; what  is  the  difference  in  longitude? 

12.  If  both  places  have  E.  longitude,  how  do  you  find 

the  difference  in  longitude?  If  both  have  W. 
longitude?  If  one  has  E.  longitude  and  the  other 
W.  longitude? 

Exercise  8 — Written. 

Harry  and  his  father  are  traveling. 

1.  Harry  is  in  one  city,   his  father  is  in  another 

located  10°  8'  45"  away;  what  is  the  difference 

in  their  time? 

(VIII-ll) 


ARITHMETIC 

2.  Another  time  Harry  was  in  Chicago  5  hr.  50  min. 

26  sec.  west  of  Greenwich  while  his  father  was 
visiting  in  Greenwich;  how  many  ®  '  "  were 
between  them? 

3.  Harry  is  in  Washington  77°  3'  45"  west  of  London ; 

if  it  is  12  o'clock  noon  in  London,  what  time 
has  Harry? 

4.  Harry  and  his  father  met  in  New  Orleans  at  6 

o'clock  one  evening;  as  their  watches  were 
stiU  keeping  the  times  of  the  places  from  which 
they  came,  Harry's  watch  showed  the  time  to  be 
7  o'clock  while  his  father's  watch  showed  4 
o'clock;  from  which  direction  did  Harry  come? 
His  father? 

5.  If  the  longitude  of  New  Orleans  is  90°  3'  15''  West; 

from  what  longitude  did  Harry  come?  His 
father? 

6.  Drill:    Find  the  difference  in  longitude  and  the 

difference  in  time: 

(a)  82°  6'  West;  12°  T  15"  East; 
(6)  100°  2'  15''  East;  25°  5'  10''  East; 
(c)   10°  V  6"  East;  4°  V  6"  West; 
\d)  105°  6'  r  West;  12°  8'  V  West; 
(e)  10°  15'  6"  East;  10°  15'  6"  West. 

7.  The  longitudes  of  two  ships  at  sea  are  respectively 

30°  12'  15"  W.  and  10°  20'  30"  E.;  what  is  the 
difference  in  their  longitude?  What  is  the 
difference  in  their  time? 

8.  When  the  meridian  on  which  your  school  is  located 

passes  under  the  sun  it  is  noon;  what  time  is 
it  then  in  a  city  located  25°  10'  15"  farther  west? 

(VIII-12) 


DENOMINATE  NUMBERS 

9.  What  difference  is  there  in  the  longitudes  of  your 
school  and  a  school  located  at  a  point  where  it 
is  3.30  p,  M.  when  you  have  noon?  Would  such 
a  school  be  east  or  west  of  vou? 
10.  What  is  the  time  and  day  in  San  Francisco,  Cali- 
fornia, 122°  25'  30"  W.  longitude  when  it  is  4 
A.  M.  of  Thursday  at  Greenwich  near  London, 
England? 


Lesson  4 
Standard  Time  in  the  United  States 


The  Four  Time  Zones— 1920 

In  1883  the  railroads  of  the  United  States  established 
a  set  schedule  which  divides  the  United  States  into  four 
time  zones  or  divisions,  known  as  follows:  Eastern 
Time,  Central  Time,  Mountain  Time,  and  Pacific 
Time.  The  heavy  zigzag  lines  separate  these  divisions 
on  the  map. 

(VIII-13) 


ARITHMETIC 

The  time  used  in  each  of  these  divisions  is  that  of 
longitude  75°  W.,  90°  W.,  105°  W.  and  120°  W.  and 
standard  or  railroad  time  is  the  same  throughout  a 
whole  division.  The  division  hues  touch  important 
railroad  terminals.  All  this  was  done  to  save  confusion 
and  danger  of  accidents  resulting  from  differences  in 
time. 

Exercise  9 — Oral. 

Harry  has  ''time  trouble"  in  the  United  States. 
Help  him. 

1.  Harry  traveled  from  coast  to  coast;  he  covered 

about  how  many  degrees? 

2.  His  watch  said  5  a.  m.  when  he  left  the  75th 

meridian;  when  should  he  change  his  watch, 
when  he  crossed  the  division  line  between  the 
two  time  zones  or  when  he  reached  the  90th 
meridian?  Should  he  move  it  back  or  ahead? 
How  much? 

3.  How  much  did  he  change  his  time  to  be  correct  if 

he  went  from  the  75th  meridian  to  the  120th? 
Did  he  set  his  watch  ahead  or  back? 

4.  AVhen  it  was  9.30  a.  m.  (Standard  Tune)  in  New 

York,  what  time  did  his  father  have  if  he  was  in 
Chicago? 

5.  When  his  father  had  3.45  p.  m.  in  Iowa,  what  time 

had  Harry  in  Delaware? 

6.  When  Harry's  father  had  12.00  midnight  in  Colo- 

rado, what  tune  had  Harry  in  Illinois? 

7.  Harry  wanted  to  find  out  what  good  there  was  in 

ha\'ing  Standard  Time;   can  you  tell  him? 

(VITI-14) 


DENOMINATE   NUMBERS 

8.  He  could  name  5  noted  cities  that  have  Eastern 

Time;  can  you? 

9.  He  could  name  5  noted  cities  that  have  Mountain 

Time;  can  you? 

10.  He  could  name  2  noted  cities  that  have  Pacific 

Time;  can  you? 

11.  He  met  his  father  in  Wyoming;     Harry's  watch 

said  2  p.  M.,  his  father's  said  noon;  if  his  father 
had  changed  his  watch  to  Standard  Time,  what 
change  must  Harry  make?  Which  way  had 
Harry  traveled?     How  many  degrees? 

12.  Harry  met  his  father  in  Chicago;  Harry  had  come 

from  eastern  Texas,  his  father  from  western 
Ohio;  who  was  ahead  by  his  watch  if  neither 
had  changed  time  on  the  trip? 

13.  Harry  knew  on  what  meridian  in  the  Eastern 

Time  Zone  sun  time  and  Standard  Time  are 
alike;  do  you  know? 

14.  He  knew  on  what  meridian  in  the  Central  Time 

Zone  sun  time  and  Standard  Time  are  alike;  do 
you  know^? 

15.  Could  you   tell  him   on   what   meridian   in   the 

Mountain  Time  Zone  sun  time  and  Standard 
Time  are  alike? 

16.  Could  you  tell  him  on  what  meridian  in  the 

Pacific  Time  Zone  sun  time  and  Standard  Time 
are  alike? 

17.  Harry  said   that   there  are  more  places  in  the 

United  States  that  use  Standard  Time  or  rail- 
road time  than  there  are  places  that  use  sun 
time;  do  you  think  he  was  right?     T\Tiy? 

(VIII-15) 


ARITHMETIC 


Lesson  5 
The  International  Date  Line 


EAST  WEST 

LONGITUDE      LONGITUDE 


ooooo    ooc* 

o  w  ->«    «>   oc     —      ■    -- 


The  180th  meridian  was 
selected  by  the  nations  of  the 
world  as  the  dividing  line  be- 
tween dates  and  between  the 
East  and  the  West  (called  the 
International  Date  Line)  be- 
cause it  marks  the  distance  half 
way  around  the  earth  from 
Greenwich,  and  also  because  it  is 
located  almost  entirely  in  the 
Pacific  Ocean,  which  avoids  the 
confusion  that  might  result  were 
a  densely  populated  location 
selected.  The  actual  date  line 
varies  some  places  from  the  180th 
meridian  to  avoid  populated  land. 

When  a  ship  crosses  the  180th  meridian  from  east 
to  west  at  noon  on  Monday,  Monday  noon  instantly 
changes  to  Tuesday  noon  (or  add  24  hr.)  and  should 
this  ship  at  once  turn  about  and  recross  from  west  to 
east,  Tuesday  noon  would  again  change  back  to 
Monday  noon  (or  subtract  24  hr.) 


The  International  Date 
Line 


Exercise  10 — Oral. 

Harry  goes  to  sea. 

1.  Harry's  ship  crosses  the  International  Date  Line 
from  west  to  east  at  11.59  p.  m.  Saturday  night 
and  continues  traveling  toward  the  east;  how 
(vni-16) 


DENOMINATE   NUMBERS 

long  will  it  have  been  Saturday  on  board  this 
ship  when  Sunday  begins? 

2.  If  Harry's  ship  crosses  the  International  Date 

Line  from  east  to  west  at  11.59  p.  m.  Saturday 
night  and  continues  traveling  toward  the  west, 
how  long  will  it  have  been  Sunday  on  board 
this  ship  when  Monday  begins? 

3.  Harry  studies  the  location  of  the  International 

Date  Line  and  wonders  why  it  is  so  irregular; 
can  you  tell  him? 

4.  When  Harry  crossed  the  International  Date  Line, 

how  far  was  he  from  the  Prime  Meridian?  How 
many  hours  difference  in  time? 

5.  What  can  you  tell  Harry  about  time  in  different 

parts  of  the  world? 

6.  Harry's  ship  crossed  the  45th   meridian    (west 

longitude)  at  10  a.  m.  Tuesday;  what  time  was  it 
then  on  each  of  the  other  meridians  shown  on 
the  map  at  the  beginning  of  Lesson  3  going 
around  both  ways  to  the  180th  meridian? 

7.  When  it  is  1  a.  m.  on  the  Prime  Meridian,  what 

time  is  it  on  each  of  the  other  meridians  shown 
on  the  map  in  Lesson  3? 

8.  When  it  is  3  p.  m.  on  the  90th  meridian   (east 

longitude),  what  time  is  it  on  each  of  the  other 
meridians  shown  on  the  map  in  Lesson  3? 

Exercise  11 — Written. 

Drill:    Find  the  difference  in  time;   then  the  dif- 
ference in  longitude : 

1.  1  hr.  2  min.  6  sec.  p.  m.  and  12  noon. 

(VIII-17) 


ARITHMETIC 

2.  2  hr.  6  min.  10  sec.  p.  m.  and  4  hr.  15  min.  20 

sec.  p.  M. 

3.  12  noon  and  4  hr.  4  min.  30  sec.  p.  m. 

4.  3  hr.  2  min.  45  sec.  a.  m.  and  12  noon. 

5.  1  A.  M.  and  6  hr.  4  min.  8  sec.  a.  m. 

6.  Change  4°  14'  8"  to  its  equivalent  in  time  units. 

7.  Reduce  4  hr.  6  min.  10  sec.  to  its  equivalent  in 

longitude. 

8.  Reduce  172°  40'  30"  to  its  equivalent  in  time  units. 

Lesson  6 
The  Metric  System 

The  ''metric  system"  is  a  decimal  system  of  weights 
and  measures  devised  and  adopted  by  the  French  in 
1789.  On  account  of  its  simplicity,  it  is  coming  more 
and  more  into  general  use.  For  scientific  and  labora- 
tory purposes  it  is  used  almost  universally. 

Since  the  "World  War  the  metric  system  has  become 
more  vital  to  the  United  States  than  ever  before; 
besides,  every  South  American  country  now  uses  this 
method  and  for  this  reason  finds  it  easier  to  trade  with 
European  countries. 

In  the  metric  system,  10  units  of  one  measure  are 
equal  to  1  unit  of  the  next  larger  measure  in  the  same 
way  as  in  our  decimal  system. 

There  are  five  units  of  measure  used  in  the  metric 
system. 

meter  (m.)  unit  of  length. 

(1  m.  =  39.37  in.) 
square  meter  (sq.  m.)  unit  of  area, 
cubic  meter  (cu.  m.)  unit  of  volume. 

(VIII-18) 


DENOMINATE   NUMBERS 


gram  (g.)  unit  of  weight. 

(1  g.  =  .035  oz.) 
liter  (1.)  unit  of  capacity. 

(1  1.  =  TT  qt.  Dry — for  grains,  etc.) 
(1  1.  =  IxV  qt.  Liquid — for  all  liquids.) 
Now,  bearing  in  mind  that  the  metric  system  is  based 
on  the  decimal  principle,  anyone  can  construct  all  of 
the  tables  used  in  the  metric  system  by  learning  these 
few  simple  Greek  and  Latin  prefixes  which  designate 
the  different  decimal  values: 


Greek  < 


f  Myria  (M.)  meaning 10,000. 

Kilo  (K.)  meaning 1,000. 


Hecto  (H.)  meaning 

100. 

,  Deka  (D.)  meaning 

10. 

Use  name  of  unit  for 

1. 

'  deci  (d.)  meaning 

.1 

Latin  <  centi  (c.)  meaning 

.01 

^  milli  (m.)  meaning 

.001 

This  gives  us  the  following  comparative  table : 


Decimal 
Scale 

10,000. 
(1,000.x  10) 

1,000. 
(lOO.XlO) 

100. 
(lO.X  10) 

10. 
(l.X  10) 

1. 
(.1  X  10) 

.1 

(.01  X  10) 

.01 
(.001  X  10) 

.001 
(-) 

U.S.Money 

— 

— 

— 

Eagle 

dollar 

dime  (d.) 

cent,  (c.) 

mill  (m.) 

Length 

Myria  (M.) 

Kilo  (K.) 

Hecto  (H.) 

Deka  (D.) 

meter  (m.) 

deci  (d.) 

centi  (c.) 

milli  (m.) 

Weight  ... 

Myria  (M.) 

Kilo  (K.) 

Hecto  (H.) 

Deka  (D.) 

gram  (g.) 

deci  (d.) 

centi  (c.) 

miUi  (m.) 

Capacity . 

Myria  (M.)" 

Kilo  (K.) 

Hecto  (H.) 

Deka  (D.) 

liter  a.) 

deci  (d.) 

centi  (c.) 

milli  (m.) 

(Note  the  similarity  between  dime  and  deci,  cent  and 
centi,  mill  and  milli  and  the  abbreviations  thereof. 
Also  note  that  the  abbreviations  for  the  terms  larger 
than  1  are  in  large  letters,  while  those  for  the  terms 
smaller  than  1  are  in  small  letters.) 

(VIII-19> 


ARITHMETIC 

The  tables  of  the  metric  system  are  constructed  by 
using  the  proper  unit  of  measure  with  the  various 
prefixes  to  show  the  proper  decimal  values,  as  follows: 

Table  of  Length  Measure 

(1  meter  =  39.37  inches) 

10  millimeters  (m.  m.) =1  centimeter  (c.  m.) 

10  centimeters =1  decimeter  (d.  m.) 

10  decimeters =1  meter  (m.) 

10  meters =1  Dekameter  (D.  m.) 

10  Dekameters =  1  Hectometer  (H.  m.) 

10  Hectometers =  1  Kilometer  (K.  m.) 

10  Kilometers =1  Myriameter  (M.  m.) 

Table  of  Weight  Measure 
(1  gram  =  .035  oz.) 

Same  as  Table  of  Length  Measure  excepting  that 
'•gram"  must  be  substituted  for  ''meter." 
Build  your  own  Table  of  Weight  Measure. 

Table  of  Capacity  Measure 

(1  liter  =    I   ^^  ^^'  ^^^^ 

I  liV  qt.  Liquid) 

Same  as  Table  of  Length  Measure  excepting  that 
''liter"  must  be  substituted  for  "meter." 
Build  your  own  Table  of  Capacity  Measure. 

Table  of  Square  Measure 
Just  as  1  square  foot  contains  144  square  inches, 
(12  in.  long  and  12  in.  wide),  so  1  square  centimeter 
contains  100  square  millimeters  (10.  m.  m.  long  and 
10  m.  m.  wide);  therefore,  100  units  of  one  square 
measure  are  equal  to  1  unit  of  the  next  larger  measure. 

(VIII-20) 


DENOMINATE   NUMBERS 


10  d.m. 


Isq.m. 


The  table  is  the  same  as  the  Table  of  Length  Measure 
excepting  that  100  must  be  substituted  for  10,  because 
each  area  is  10  units  square,  and  the 
word  '^square"  must  be  prefixed  to 
each   measure. 

100  square  millimeters  (sq.  m.  m.) 
=  1  square  centimeter  (sq.  c.  m.). 

Build    your  own   Table    of   Square 
Measure.  ^ 

Table  of  Cubic  Measure 
Just  as  1  cubic  foot  contains  1,728  cubic  inches, 
(12  in.  long,  12  in.  wide  and  12  in.  thick),  so  1  cubic 
centimeter  contains  1,000  cubic  millimeters  (10  m.  m. 
long,  10  m.  m.  wide  and  10  m.  m.  thick);  therefore, 
1,000  units  of  one  cubic  measure  are  equal  to  1  unit  of 
the  next  larger  measure. 

The  table  is  the  same  as  the  Table  of  Length  Measure 
excepting  that  1,000  must  be  sub- 
stituted for  10,  and  the  word  ^^ cubic'' 
must  be  prefixed  to  each  measure. 

1,000  cubic  millimeters  (cu.  m.  m.) 
=  1  cubic  centimeter  (cu.  cm.) 
Build    your  own  Table    of   Cubic 
1000  cud Tn.=  i'^uin.  Measure. 


Exercise  12 — Oral. 

Children  are  to  ask  each  other: 

1.  What  prefix  means  1,000?     100?     10,000?     10? 

.001?     .1?     .01? 

2.  What  are  the  abbreviations  for  each  of  the  prefixes 

in  Question  1? 

(VIII-21) 


ARITHMETIC 

3.  Which  prefixes  have  large  letters  for  their  abbrevi- 

ations?    Which  have  small  letters? 

4.  In  the  metric  system  Table  of  Length  Measure, 

how  many  units  of  one  measure  are  equal  to  1 
unit  of  the  next  larger  measure?  What  is  the 
unit  of  length  called? 

5.  In  the  metric  system  Table  of  Weight  Measure, 

how  many  units  of  one  measure  are  equal  to  1 
unit  of  the  next  larger  measure?  What  is  the 
unit  of  weight  called? 

6.  In  the  metric  system  Table  of  Capacity  Measure, 

how  many  units  of  one  measure  are  equal  to  1 
unit  of  the  next  larger  measure?  What  is  the 
unit  of  capacity  called? 

7.  In  the  metric  system  Table  of  Square  Measure, 

how  many  units  of  one  measure  are  equal  to  1 
unit  of  the  next  larger  measure?  Why  is  this 
right?     What  is  the  unit  of  area  called? 

8.  In  the  metric  system  Table  of  Cubic  Measure, 

how  many  units  of  one  measure  are  equal  to  1 
unit  of  the  next  larger  measure?  Why  so  many? 
What  is  the  unit  of  volume  called? 

What  is  equal  to: 

9.  10  millimeters?  17.  10  milligrams? 

10.  10  centigrams?  18.  10  centimeters? 

11.  10  deciliters  19.  1,000  m.  m.? 

12.  100  square  meters?  20.  100  K.  m.? 

13.  1,000  cubic  Dekameters?      21.  100  d.  g.? 

14.  10  Kilometers?  22.  1,000  cu.  d.  m.? 

15.  10  Dekagrams?  23.  100  sq.  D.  m.? 

16.  10  Hectoliters?  24.  10  centiliters? 

(VIII-22) 


DENOMINATE   NUMBERS 

Exercise  13 — Written. 

Name  the  fundamental  unit  of  each.  Arrange  in 
order  and  add : 

1.  6  m.  m.,     3  c.  m.,     9  D.  m.,     7  H.  m.,     1  K.  m. 

2.  2  c.  g.,    20  d.  g.,    55  D.  g.,    9  D.  g.,     15  c.  g. 

3.  20  1.,     15  d.  1.,     25  c.  1.,     9  K.  1.,     15  H.  1.,     6  1. 

4.  Find  cost  of  No.  3  if  each  Uter  costs  28^. 

5.  Find  value  of  #2  if  each  gram  costs  $.05. 

6.  If  a  box  of  biscuit  weighs  340  g.  what  will  a  gross 

of  boxes  weigh?     (Answer  in  grams.) 

7.  Answer  No.  6  in  oz.     In  lb. 

Lesson  7 
Foreign  Money 

In  order  that  we  may  have  business  dealings  with 
people  in  foreign  countries,  it  is  necessary  that  we  know 
the  value  of  the  different  denominations  of  money  used 
in  such  countries. 

Table  of  English  Money 

12  pence  (d.). .  =  1  shilling  (s.)..  =  $0,243+ 
20  shillings. . . .  =  1  pound  (£)...  =    4.8665 

Table  of  French  Money 

100  centimes  (c.)  =  1  franc  (fr.)  =  $0,193 

Money  used  in  other  European  Countries 

A  Belgian  franc    "> 


1.193  each. 


A  Spanish  peseta 
An  Italian  lira 
A  Swiss  franc 
(Each  of  these  has  the  same  value  as  the  French  franc.) 

(VIII-23) 


ARITHMETIC 

You  may  be  interested  to  know  the  value  of  the 
monetary  unit  used  in  some  of  the  other  principal  coun- 
tries of  the  world;  these  you  need  not  learn,  but  you 
can  use  them  for  reference : 

Country  Monetary  Unit  U.  S.  Value 

Canada dollar $1.00 

Germany mark 0.2382 

Japan yen 0.4985 

Norway crown 0.268 

Sweden crown 0.268 

South  American  Countries: 

Argentine  Republic peso 0.9648 

Bolivia boliviano 0.3893 

Brazil milreis 0.5462 

Chile peso 0.365 

Peru libra 4.8665 

Venezuela bolivar 0.193 

Exercise  14 — Written. 

Study  first  to  decide  which  is  the  best  way  of  doing 
these. 

Find  the  value  of  the  following  in  United  States 
money : 

1.  £50 

2.  £4  16s. 

3.  £2  6d. 

4.  5s.  lOd. 

5.  £400  19s.  2d. 

6.  50fr. 

7.  75c. 

8.  68fr.  30c. 

(VIII-24) 


>  Time  yourself. 


DENOMINATE   NUMBERS 


Find  the  value  of  the  following  in  English  money: 
9.        $29.20^ 

>  Time  yourself. 


10.  $245.08 

11.  $418.58 

12.  $2,433.25  J 


Find  the  value  of  the  following  in  United  States 
money : 

13.  175  Belgian  francs.  ' 

14.  300  Italian  lira. 

15.  80  Spanish  peseta.  \  Time  yourself. 

16.  450  Swiss  francs. 

17.  290  French  francs. 

Find  the  value  of  the  following: 

18.  $14.47  in  Italian  lira. 

19.  $82.99  in  Spanish  peseta. 

20.  $11.58  in  Swiss  francs. 

21.  $96.50  in  French  francs. 

22.  $17,37  in  Belgian  francs. 


Time  yourself. 


Exercise  15 — Written. 

A  Trip  through  Europe 

John  Walker's  father  is  a  buyer  for  Marshall  &  Co.^s 
large  store  and  goes  abroad  every  summer  on  business. 
Last  summer  he  took  John  along.  ^' Watch  your  time 
and  money,  John,"  was  his  father's  advice. 

1.  They  left  Chicago,  111.,  at  12.40  p.  m.,  on  one  of 
the  fast  trains  which  reaches  New  York  in  20 
hours;  what  time  of  the  day  did  this  train 
arrive  in  New  York? 

(VIII-25) 


ARITHMETIC 

2.  They  sailed  July  8th  from  New  York  City  on  a 

British  ship  bound  for  Southampton.  While  on 
board  ship,  Mr.  Walker  paid  £5  10s.  for  mis- 
cellaneous expenses;  how  much  was  this  in 
United  States  money? 

3.  John,  who  had  set  his  watch  properly  in  New 

York  City,  made  no  change  in  his  time  while  on 
the  ocean,  but  his  father  set  his  watch  several 
times  on  the  way  so  that  it  agreed  with  London 
time  on  their  arrival  at  Southampton.  Wliat 
was  the  difference  in  time  as  shown  by  these  two 
watches?  Did  John  have  to  set  his  watch  for- 
ward or  backward  to  correct  it? 

4.  They  immediately  went  by  train  from  South- 

ampton to  London,  to  which  point  they  had 
bought  tickets  in  Chicago.  Arriving  in  London, 
they  took  a  taxicab  to  the  hotel,  paying  the 
chauffeur  10s.  6d.;  how  much  was  this  in 
United  States  money? 

5.  The  next  day  Mr.  Walker  bought  the  following 

invoice  of  Nottingham  lace : 

500  yards  2"  Lace  @  5s.  6d. 
800  yards  3"  Lace  @  7s. 
1,000  yards  4"  Lace  @  10s. 

How  much  did  each  item  on  this  bill  amount  to 
in  United  States  money?  How  much  did  the 
total  bill  amount  to  in  United  States  money? 

6.  That  evening  they  settled  their  hotel  bill  for  two 

days  at  15s.  per  day  per  person,  and  left  for 
Brussels,  Belgium;  what  was  the  amount  of  the 
hotel  bill  in  United  States  money? 

(VIII-26) 


DENOMINATE   NUMBERS 

7.  At  Brussels,   Mr.   Walker  bought   the  following 

Belgian  lace: 

6  Hectometers .  .  @  5  fr.  per  m. 
1  Kilometer  ....  @  6  fr.  per  m. 

What  was  the  total  amount  of  this  bill  in  United 
States  money? 

8.  When  getting  ready  to  leave  Brussels  for  Paris, 

Mr.  Walker  arranged  for  transportation  through 
the  hotel  clerk  and  found  his  entire  bill  amounted 
to  80  fr.  Having  no  Belgian  money,  he  handed 
the  clerk  £5  in  English  money;  how  many  francs 
and  centimes  did  he  receive  as  change? 

9.  After  making  a  number  of  purchases  at  Paris,  Mr. 

Walker  decided  they  would  spend  ten  days  sight- 
seeing in  France,  Switzerland,  Spain  and  Italy 
before  sailing  for  home.  Their  entire  expenses 
for  these  ten  days  were  as  follows : 

In  France 223  francs; 

In  Switzerland 200  francs ; 

In  Spain 152  peseta; 

In  Italy 175  lira; 

How  much  was  this  in  United  States  money? 

10.  They  bought  tickets  from  Italy  direct  to  Chicago 

for  2,000  lira;  how  much  was  this  in  United 
States  money? 

11.  In  Rome,  Italy,  John  had  set  his  watch  according 

to  the  time  of  longitude  15°  E.  and  he  did  not 
change  it  until  he  reached  Chicago;  what  had 
he  to  do  to  correct  his  watch  to  Standard  Central 
Time? 

(yiU-27) 


ARITHMETIC 

Exercise  16 — Oral  Review. 

Use  your  eyes  quickly — keep  looking  ahead. 

1.  Read: 

(a)  986,438,721,863,410; 
(6)  876,439,000,000,001; 
(c)     38,000,468,000,612. 

2.  Add  8,000,000;  9,000,000;  17,000,000;  20,000,000. 

3.  Add  6i;  13^;  10|;  5|. 

4.  Add  .25;  .75;  .75;  .75;  .25; 

5.  Add66f%;  66|%;  33J%;  33|%;  66|%. 

6.  Give  the  sum  of  J;  |;  f;  I;   f;   f;  J;   |. 

7.  Define  longitude. 

8.  How  much  time  corresponds  to  360°  of  longitude? 

90''?     15°?     1°?     r?     1''? 

9.  Name  the  metric  system  unit  used  in  measuring 

ribbons.     In  measuring  the  area  of  a  lot.     The 

amount  of  grain  in  a  bin.     The  weight  of  a  car. 

10.  What  single  word  means  10,000  meters?     1,000 

m.?     100  1.?     10  g.?     .1  m.?     .01  g.?     .001  1.? 

Exercise  17 — Written  Review. 

1.  Write: 

(a)  Four  hundred  sixteen  trillion,  eight  thousand, 

sixteen. 
(h)  Thirty-eight  billion,  one  hundred  sixty-four 

million,  nine  thousand,  twelve, 
(c)   Fifty-eight  trillion,  six  million,  six. 

2.  Reduce: 

(a)  36,410"  to  highest  terms; 

(b)  86°30'' to  seconds  of  arc; 

(c)  90°  40'  20"  to  minutes  of  arc. 

^       (VIII-28) 


DENOMINATE   NUMBERS 

3.  Two  cities  are  15°  15'  15"  apart;    what  is   the 

difference  in  their  local  time? 

4.  The  longitudes  of  two  cities  are  respectively:   95° 

50'  35"  E.  and  65°  30'  40"  W.  What  is  the 
difference  in  their  longitude?  What  is  the 
difference  in  their  time? 

5.  What  time  is  it  45°  30'  east  of  you  when  it  is  11 

p.  M.  in  your  locality? 

6.  What  time  is  it  75°  30"  west  of  you  when  it  is  1 

A.  M.  in  your  locahty? 

7.  Add  684.32;    25.684;    695.404;    2.6568;    .7923; 

4.0005;  7. 

8.  From  12,000,000  take  8,654,209J. 

9.  Find  the  compound  mterest  on  $3,000.00  for  1| 

years  at  6%  payable  semi-annually. 
10.  At  what  rate  per  cent  will  $76.00  amount   to 
$77.52  in  120  days? 

Subtract,  but  do  not  copy: 

(Time  for  these  12  examples  is  less  than  4  minutes.) 
11.  12.  13.  14. 

386,419  741,038  563,875  619,024 

289,523  258,063  498,987  285,387 

15.  16.  17.  18. 

400,004    738,056    293,712    611,041 
193,708    295,386    196,874    298,309 


19.        20.        21.        22. 

901,315    612,041    814,002  619,312 

198,749    387,592    319,704  412,961 

(VIII-29) 


ARITHMETIC 

Copy  and  multiply: 

(Time  for  these  8  examples  is  less  than  4  minutes.) 


23.  2,587  X  53 

24.  9,342  X  82 

25.  7,624  X  49 

26.  8,041  X  67 


27.  1,758  X  34 

28.  5,923  X  75 

29.  4,328  X  92 

30.  6,407  X  86. 


Copy  and  divide: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 

31.  14,246  -T-  419;  34.  36,146  -r-  682; 

32.  33,087  -T-  807;  35.  19,351  ^  523; 

33.  37,440  -^  720;  36.  20,736  -r-  324. 

Add,  but  do  not  copy: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 


37. 

38. 

39. 

40. 

41. 

42. 

7,341 

6,194 

1,234 

1,787 

2,631 

4,879 

8,629 

9,387 

5,874 

8,641 

1,649 

3,092 

6,295 

8,495 

9,021 

3,953 

9,345 

2,877 

3,129 

4,729 

3,849 

7,865 

5,076 

8,985 

4,728 

6,028 

6,128 

2,013 

6,032 

6,139 

5,138 

4,197 

2,941 

8,741 

3,775 

8,229 

(VIII-30) 


POWERS   AND   ROOTS 

Lesson  8 

What  Powers  Are — Squaring  and  Cubing 

The  product  obtained  by  using  any  number  several 
times  as  a  factor  is  a  ''power"  of  that  number.     Thus: 

5  used  twice  as  a  factor  gives  us  the  2d  power  of 

5,  or  25  (5  X  5). 
5  used  three  times  as  a  factor  gives  us  the  3d  power 

of  5,  or  125  (5  X  5  X  5). 

The  2d  power  of  any  number  is  called  the  ''square" 
of  that  number;  the  3d  power  is  called  the  ''cube"  of 
that  number. 

5  X  5  =  25,  also  written  5  square,  or  5^. 
5X5X5  =  125,  also  written  5  cube,  or  5^ 

The  small  figure  telling  the  power  of  the  number  is 
called  the  exponent.    What  exponents  were  used  here? 

Exercise  18 — Oral. 

1.  Give  the  areas  of  the  following: 


4. 

9 

9 

1 

6 

6 

10 

A. 

10 


2.  4'  =  ?;  9'  =  ?;   l'  =  ?;  6'  =  ?;   10'  =  ?;  3'  =  ?. 

3.  Give  the  second  power  of  5;  4;  7;  8;  6;  a;  x. 

4.  Give  the  square  of  12;  c;  3;  a;  4. 


(VIII-31) 


ARITHMETIC 


5.  Give  the  volumes  of  the  following: 


19. 


io 


^         5 

6        - 

pn 

> 

5 

1  1 

(3^ 

x' 

IO 


a- 


6.  2'  =  ?;  5'  =  ?;  1'  =  ?;  6^  =  ?;  lO'^  =  ?;  3'  =  ?. 

7.  Give  the  third  power  of  3;  2;  6;  5;  4;  1. 

8.  Give  the  cube  of  2;  m;  4;  1;  c. 

9.  Read  the  exponents  here:  4^;   m^;  4^•   7^   8^;  2\ 

10.  Read  and  find  the  values  of:  2^•  (\f\  (i)^•  12^;  .2^; 

11.  By  cross  multipHcation  find  the  square  of  any 

number  composed  of  2  digits. 

Example:  23^  =  ^>^  3X3=     9       1st  step, 

^i^  -^         2(3  X  2)  =  12       2d  step. 
"529       (2  X  2)  +  1  =    5      3d  step. 


Exercise  19 — Written. 
Find  the  squares  of: 


1.    15; 

6. 

53. 

;      11. 

1. 

2> 

16.  8i; 

2.      4i; 

7. 

81 

;      12. 

3. 
4; 

17.  3i; 

3.    35; 

8. 

75, 

13. 

5  . 
2  J 

18.  d\ 

4.  125; 

9. 

26, 

14. 

JLi. 

12, 

19.  x; 

5.  400; 

10. 

.82 

I;       15. 

02; 

20.  a. 

ad  the  cubes  of: 

21.     32; 

25. 

21 

;        29. 

3. 

7  ? 

33.  2f ; 

22.     45; 

26. 

2 

1;      30. 

.25; 

34.  1|; 

23.       9.6; 

27. 

64 

;        31. 

/t; 

35.  c; 

24.  100; 

28. 

48 
( 

;        32. 

VIII-32) 

2. 
3> 

36.  m. 

POWERS  AND   ROOTS 

Find  the  values  of: 

37.  2.5^•         38.  .12^•       39.  I.IS^;       40.  (f)l 

Lesson  9 

What  Roots  Are 

Any  one  of  the  equal  factors  which  produce  a  number 
IS  called  a  '^root"  of  that  number;  thus  a  number  is 
the  ''root"  of  all  of  its  "powers." 

The  root  of  a  2d  power  is  called  a  ''square  root"; 
therefore,  since  5X5^  25,  the  square  root  of  25  is  5. 
5  is  one  of  the  two  equal  factors  that  produce  25. 

The  root  of  a  3d  power  is  called  a  "cube  root"; 
therefore,  since  5X5X5  =  125,  the  ;  cube  root  of 
125  is  5.  5  is  one  of  the  three  equal  factors  that 
produce  125. 

To  indicate  square  root,  the  symbol  V  called  a 

"radical  sign"  is  used;  thus,  \/25  =  5.  A  small 
figure  ^  called  an  "index"  is  sometimes  written  in  the 
radical  sign  to  indicate  square  root;  thus,  ^^25  =  5, 
but  this  is  unnecessary  as  square  root  is  the  simplest 
form  of  root,  and  square  root  is  understood  unless 
some  other  index  is  used,  as : 

(a)   ^25  means  square  root  of  25.     Ans.,  5. 

(6)   >^125  means  cube  root  of  125.     Ans.,  5. 

Fxercise  20 — Oral. 

1.  Wliat  is  the  square  root  of  4?  Of  49?  Of  144?  Of  J? 

2.  WTiat  is  the  cube  root  of  8?    Of  64?    Of  27^    Off? 

3.  If  the  cube  is  8,  what  is  the  root  from  which  it 

came? 

(VIII-33) 


ARITHMETIC 

4.  V36  =  6;    explain  which  is  the  power.      Which 

is  the  root? 

5.  yl27  =  3;   explain  which  is  the  root.      Which  is 

the  power? 

6.  Indicate  that  the  square  root  of  169  is  to  be 

extracted.     Indicate  that  the  cube  root  of  m^ 
is  to  be  extracted. 

7.  What  name  is  given  to  the  root  sign? 

8.  Read,  and  find  the  values  of: 

V8l  =  ?     ^27  =  ?     \'T4i  =  ?     ^8  =  ? 
V.64  =  ?      ^1  =  ? 

9.  Read  the  index  figures  in  Question  8. 
10.  Read  the  powers  in  Question  8. 

Lesson  10 
How  to  Extract  the  Square  Root  (Integers) 

As  you  have  already  learned,  the  square  of  a  number 
is  the  product  obtained  by  using  that  number  twice  as 
a  factor;  hence,  to  find  the  square  root  when  we  know 
the  square,  we  must  reverse  the  process  and  find  the 
number  which  was  used  twice  as  a  factor  to  produce 
such  square. 

Root    Square      Root  Square  Root  Square 

P=    1;         102=      100;        100^=    10,000; 
92  =  81;        99^  =  9,801;        999^-998,001. 

By  looking  over  the  table  here  given  which  shows  the 
smallest  and  largest  numbers  containing  one,  two,  and 
three  digits,  you  will  notice  that  when  a  number  com- 
posed of  one  digit  is  squared,  the  square  contains  either 
one  or  two  digits;    when  a  number  composed  of  two 

(VIII-34) 


POWERS   AND   ROOTS 

digits  is  squared,  the  square  contains  either  three  or 
four  digits;  and  when  a  number  composed  of  three 
digits  is  squared,  the  square  contains  either  five  or  six 
digits;  so  if  we  allow  2  places  for  each  digit  in  the  root 
we  will  certainly  have  enough  for  even  the  highest; 
therefore,  by  beginning  at  the  units  always  and 
separating  a  number  into  periods  of  two  digits  each, 
we  know  that  the  square  root  will  contain  as  many 
digits  as  there  are  periods  in  the  number. 


Exercise  21 — Oral. 

For  practice,  tell  at  sight  how  many  figures  there  will 
be  in  the  square  root  of  each  of  the  following  numbers: 


1. 
2. 
3. 

4. 
5. 


144 
169 

529 
6,561 

6,889 


6.  1,681 

7.  900 

8.  289 

9.  625 
10.  2,500 


11.  1,024 

12.  8,281 

13.  3,844 

14.  5,625 

15.  7,225 


Separate  the  following  numbers  into  periods  to  show 
how  many  figures  the  square  roots  will  contain: 


16.  144 

17.  625 

18.  4,900 

19.  15,625 

20.  90,601 

21.  93,636 

22.  169 


23. 
24. 

25. 
26. 
27. 
28. 
29. 


529; 

400; 
1,225; 
7,225; 
9,801 ; 
6,889; 

324; 


30.  10,000 


31. 
32. 
33. 
34. 
35. 


56,874 

4,168 

137 

96 

5 


36.   15,498. 


In  working  an  example  by  cross  multiplication,  we 
first  multiply  the  units;  second,  we  find  the  sum  of  the 
units  multiplied  by  the  tens  plus  the  tens  multiplied  by 

(VIII-35) 


ARITHMETIC 

the  units;    third,  we  multiply  the  tens;    hence,  when 
the  multipHer  and  the  multiplicand 
are  exactly  alike,  as  they  always  are     ^'"^^^^  *^^  ^^"^'^  ^^ 
in  squaring  a  number,  the  first  step      ^^^^^  Multiplication 
is  equivalent  to  squaring  the  units,    First  Step      13^ 
3^  =  9;  the  second  step  is  equiva- 
lent to  finding  the  product  of  twice 
the   tens   multiplied    hy    the    units, 
2  (1  X  3)  =  6;   and  the  third  step 
is  equivalent  to  squaring  the  tens, 
P=  1. 


_13^ 
9 
Second  Step  13 
13 
69 
Thh-d  Step|^13 
V13 


169 
All  the  parts  in  the  square  of  13  =  169. 


lO 


+     3 


AREM.     lOO 


lO 


tXTJL 

TOTAL 

(lOX3  =  30) 


t  XTl 

UL^ 

(lO  X3  =30) 

(3^=9) 

lO 

3 

t^ 

/^ 

6 

0)      X 

o 

(lO^=  lOO) 

<1   p 

0 

lO  3 


AREA 


t  xu 
=  69 

(lO  X3  =  30) 


■UL' 


(3^=9)1 


(10  X  2)  +  3  =  length 
3  =  width 
23  X  3  =  area  69 

(VIII-36) 


PO\A^RS  AND   ROOTS 

Reversing  this  process,  we  separate  the  square  (169) 
into  periods  of  two  figures  each,  and  starting  with  the 
reverse  of  the  third  step,  we  find  the  greatest  square  in 
the  left  hand  period  (1  hundred)  from  which  we  know 
the  tens'  figure  of  the  root  is  1;  we  then  subtract  the 
1  hundred  and  bring  down  the  next  period  (69)  which 
resulted  originally  from  the  second  and  first  steps  of 
the  multiplication,  and,  therefore,  contains  twice  the 
tens  multiplied  by  the  units  (cross  multiplication)  plus 
the  square  of  the  units ;  w^e  therefore  divide  69  by  twice 
the  tens  (20)  and  find  this  quotient  to  be  3 ;  therefore, 
3  is  the  units'  figure,  but  before  we  complete  the  second 
step,  we  add  the  units'  figure  (3)  to  20  giving  us  23, 
so  that  when  we  multiply  by  3  w^e  have  also  reversed 
the  first  step,  since  the  3  which  w^e  added  to  20  is 
squared  when  23  is  multiplied  by  3. 


Cross  Multiplication  Reversed 
for  Finding  the  Square  Root 

1    3,  Ans. 


t  X  u 

U^ 

•o 
100           5 

X 

or 


V 


10  X  2  =  20  (2  tens) 
3  (units) 
23    X3  = 


1'69  (1  ten 
100 

69  (3  units 
(width) 

69 


lO 


=  2l 


Area  4=  69 


+ 


Trial  Divisor  =  20 
Real  Divisor  =  23  because  23  is  the  real  length, 

rviii-37) 


ARITHMETIC 


Exercise  22 — Written. 

Find  the  square  root  of  each  of  the  following  and 


e: 
1. 

169^ 

5.      625;                9.  1,024 

2. 

144 

6.  1,681;               10.  6,889 

3. 

529 

7.  4,900;               11.  1,225 

4. 

196 

8.      324;               12.  4,489 

13. 

3,136;            14.  9,604. 
Lesson  11 

How  to  Extract  the  Square  Root  (Decimals  and 

Fractions) 


EXAMPLE:     V2|  =  ? 

V2|  =  V^5.  or  f ,  or  If,  Ans. 
Proof:  (lf)2  =  2| 

Since  both  terms  of  -^/-  are  perfect 
squares,  we  can  extract  the  square 
root  of  each  term  to  find  the  cor- 
responding term  of  the  root. 

In  finding  the  square  root  of  a  fraction  we  must  find 
the  root  of  the  numerator  and  of  the  denominator 
separately,  and  this  we  cannot  do  unless  each  of  the 
two  terms  of  the  fraction  is  a  perfect  square. 

When  either  the  numerator  or  the  denominator  is 
not  a  perfect  square,  the  fraction  must  be  reduced  to 
a  decimal  before  finding  the  root,  in  which  case  the 
root  will  be  in  the  form  of  a  decimal. 

Remember,  we  can  extract  the  square  root  of  a 
common  fraction  (as  such)  only  when  each  of  its  two 
terms  is  a  perfect  square,  otherwise  the  common 
fraction  must  be  reduced  to  a  decimal  before  we  can 

extract  the  root. 

(VIII-38) 


POWERS   AND   ROOTS 


EXAMPLE:     V  1,500.  =  ? 
3  8.    7   2+,Ans. 


32  = 

15  00.0000  (3 
9 

30  X  2  =  60 
8 
68  X8  = 

6  00            (8 
5  44 

38.0  X  2  =  76.0 
.7 
76.7  X  .7  = 

56.00      (7 
53.69 

38.70  X  2  =  77.40 
.02 
77.42  X  .02  = 

2.31  00  (2 
1.54  84              ] 

.76  16  Re-      i 
mainder.      ' 

Proof:  38.722  =  1,499.2384 
Plus  Remainder  .7616 

Ooo 


We  annex  a  period 
of  two  ciphers  for 
every  decimal 
place  we  wish  to 
use  in  the  root  and 
proceed  as  before. 
Each  new  trial  di- 
visor is  obtained 
by  annexing  a  ci- 
pher to  the  root 
figures  previously 
found  (as  if  they 
were  tens),  and 
multiplying  by  2. 


In  finding  the  square  root  of  numbers  containing 
decimals,  the  periods  in  the  integer  are  separated  toward 
the  left  beginning  with  units,  and  the  periods  in  the 
decimal  are  separated  toward  the  right  from  units;  as, 

6823.4550;  an  even  number  of  decimal  places  is  always 
necessary  so  that  the  denominator  of  the  decimal  frac- 
tion may  be  a  perfect  square,  otherwise  the  denominator 
in  the  root  cannot  be  expressed  decimally;  as  we  cannot 
express  the  root  of  thousandths  decimally,  we  annexed 
a  cipher  to  .455  making  it  .4550,  and  the  root  of  ten- 
thousandths  is  hundredths. 

Exercise  23 — Oral. 

1.  Separate  the  following  numbers  into  periods  of 
two  figures  each  in  the  manner  necessary  for 

(\lII-39) 


ARITHMETIC 

extracting  the  square  root,  and  read  the  figures 
in  each  period. 

(a)  468,736; 

(b)  1,637,462; 

(c)  8,746.3621; 

(d)  96,386.472; 

2.  How  many  figures  will  there  be  in  the  square  root 

of  a  number  containing  1  or  2  figures?  3  or  4 
figures?     5  or  6  figures? 

3.  How  many  decimal  places  are  there  in  the  square 

of  .9?  Of  .19?  Of  .09?  Of  .123?  How  many 
of  these  squares  contain  an  odd  number  of 
decimal  places?  How  many  of  these  squares 
contain  an  even  number  of  decimal  places? 

4.  Why  does  the  square  of  a  decimal  always  contain 

an  even  number  of  decimal  places? 

5.  If  you  were  asked  to  extract  the  square  root  of 

.463  (.463  contains  an  odd  number  of  decimal 
places),  how  would  you  proceed  so  that  you 
could  point  off  properly  in  the  root? 

6.  How  do  we  find  the  square  root  of  f|?     Of  If  ? 

7.  Which  method  would  you  use  to  find  the  square 

root  off?    Off?     Oft?    Of  J? 

8.  Without  using  paper,  what  do  you  think  is  the 

square  root  of  169;  290;  6,400;  1;  600. 

Exercise  24 — Written. 

Find  the  square  roots  of  the  following  and  prove  your 
answers : 

1.  225.;  3.     15,625.; 

2.  1,225.;  4.       2,809.; 

(vni-40) 


POWERS  AND   ROOTS 

5.670,761.;  8.  2.;     (carry  3  places) 

6.  1.3924;        9.  1,000.; 

7.  15.625;        10.        45.; 

Exercise  25 — Written. 

Find  the  square  roots  of  the  following  and  prove  your 


answers : 


1.     IO9,  ^.      100;  «3»     J-2;  ^«       16    ;  *?•     9« 

6.  Find  the  length  of  the  side  of  a  square  whose  area 

is  289  sq.  in. 

7.  The  floor  of  a  square  room  contains  324  sq.  ft.; 

how  many  yards  long  and  wide  is  it? 

8.  What  is  the  length  of  the  perimeter  of  a  square 

containing  30|  sq.  yd.? 

9.  How  many  boards  6"  wide  will  be  needed  to  build 

a  fence  around  a  square  lot  containing  15,876 
sq.  ft.? 
10.  What  is  the  volume  (in  cubic  inches)  of  a  cube  if 
the  area  of  one  of  its  faces  is  676  sq.  in.? 

Lesson  12 
How  to  Extract  the  Square  Root  by  Factoring 


EXAMPLE:     V6,561  =  ? 
^2.  =  other  factor  82.  (Quotient)  1  Unequal 


80.)  6561.  _80.  (Divisor)    J  Factors 

640  Proof  by  2 )  162. 

161  Division               81.  (Average  Factor) 

160  81.  (The  quotient  (81)  being  the 

1  81.)  6,561.  same  as  the  divisor  (81),  we 

-Approximately  6  48  know  that  we  have  found  the 

this    seems    to  81  square  root.) 

be  one  factor.  81 


(VIII^l) 


ARITHMETIC 

Find  V6,561  by  approximating  one  factor  then 
dividing  to  find  the  other  approximately.  Of  course, 
they  will  not  be  equal  factors  until  you  average  them. 


EXAMPLE:     V2,401  =  ? 

40 

40  (Quotient   \  Unequal  Factors. 
60  (Divisor)     J 

60)2401 

2400 

2)100 

1 

50  (Average  Factor) 

Proof  by  Division 

48 

48  (Quotient)  "l  jj^        i  Factors. 
50  (Divisor)    / 

50)2401 

200 

2)98 

401 

49  (New  Average  Factor) 

400 
1 
Proof  by  Divi 

sion 

49 

49)2401 

(Now  the  quotient  (49)  is  the  same  as 

196 

as  the  divisor  (49),  so  we  know  that 

441 

we  have  found  the  square  root.) 

441 

If  your  proof  by  division  gives  a  quotient  which  is 
different  than  the  divisor,  you,  of  course,  know  that 
you  have  not  yet  found  the  square  root.  In  that  case 
use  your  divisor  and  your  quotient  as  two  new  factors, 
the  average  of  which  will  give  you  a  closer  approxima- 
tion to  the  actual  square  root.  You  can  then  prove 
again  by  division  and  if  necessary  average  again  till 
your  quotient  is  the  same  as  your  divisor. 

In  extracting  the  square  root  of  a  number  which,  is 
not  a  perfect  square,  we  can  carry  out  the  work  by 
this  method  to  any  number  of  decimal  places  that 
may  be  desired. 

(VIII-42) 


\ 


POWERS  AND   ROOTS 


Exercise  26 — Written. 

Find  the  square  roots  by  approximating  factors  and 
prove : 


1.  6,889 

6.  3,136^ 

11. 

3; 

2.  4,489 

7.  9,604 

;            12. 

2; 

3.  1,681 

8.      529 

;            13. 

5; 

4.  1,024 

9.     6.25, 

14. 

10; 

5.  1,225 

10.  12.25 

15. 

12. 

You  may  lil< 

:e  to  use 

)  this  method  when  your  numbers 

are  small. 

Exercise  27 — Written  Review. 

Add  in  five  < 

Dr  six  minutes;  then  prove: 

1. 

2. 

3. 

4. 

5. 

47,312        2 

»6,872 

1,312 

72,164 

63,128 

8,642 

4,120 

16,198 

3,190 

4,386 

97,628        c 

10,000 

8,888 

501 

72,101 

7,463 

3,612 

7,212 

83,120 

99,999 

87,211         1 

2,762 

46,328 

1,648 

1,687 

16,411        3 

6,812 

5,005 

48,729 

31,012 

Subtract  in  J 

■our  minutes;  then  prove: 

6. 

7. 

8. 

9. 

10. 

13,724        3 

6,874 

35,687 

46,342 

18,234 

1,987 

1,498 

2,129 

39,717 

306 

Multiply  in  four  minutes ;  then  prove : 

11.  4,621  X    13;  13.     138  X    75; 

12.  19,872  X  312;  14.  7,623  X  126; 

15.  1,201  X  52. 
(Villus) 


ARITHMETIC 

Divide  in  four  minutes;  then  prove: 

16.  38,645  -T-  28;  18.  41,288  -r-    8; 

17.  13,608  -^  54;  19.  13,012  -^  18; 

20.  50,000  ^  55. 
Work  in  four  minutes;  then  prove: 

21.  22.  23. 

V670,761  VT^5  Vl.3924 


(VIII-44) 


\ 


EQUATIONS 
Lesson  13 

Numbers  and  Quantities  Represented  by  Letters 

Very  often  it  is  convenient,  and  frequently  it  is 
necessary,  to  use  a  letter  to  represent  an  unknown 
number  or  quantity  while  solving  a  problem;  in  such 
cases,  when  we  have  found  the  value  of  the  letter,  we 
have,  of  course,  found  the  value  of  the  number  or 
quantity  which  that  particular  letter  represents. 

As  an  example  we  will  say  that 
a  grocer  had  a  certain  number  of 
pounds  of  sugar  on  a  scale  and 
by  placing  a  5-pound  weight  with 
it,  a  15-pound  weight  was  re- 
quired to  balance  the  scale.  Now, 
if  we  let  X  represent  the  un- 
known  number    of    pounds  of    sugar    we   find   that 

a:  -f-  5  =  15. 

Can  you  find  the  value  of  a;?     Try. 

A  statement  in  the  form  of  this  one  ^'x  +  5  =  15'^ 
is  called  an  ^'equation,''  for,  as  you  know,  an  equation 
is  a  statement  showing  the  equality  of  two  quantities 
by  placing  one  before  and  one  after  the  equality  sign. 

The  quantity  ^Titten  before  or  to  the  left  of  the 
equality  sign  is  called  the  ^^  first  member '^  of  the  equa- 
tion, and  the  one  written  after  or  to  the  right  of  the 
equality  sign  is  called  the  ^'second  member"  of  the 

(VIII-45) 


ARITHMETIC 

equation.   We  usually  use  the  last  three  letters  {x  y  z) 
of    the    alphabet   to   represent    unknown   quantities. 
Finding  the  value  of  the  letter  which  represents  the 
unknown  quantity  is  called  '' solving"  the  equation. 

Exercise  28 — Oral. 

1.  In  the  equation  y  -\-  \Q  —  12,  is  2/  a  known  or  an 

unknown  quantity?     What  is  10?     What  is  12? 

2.  Read  this  equation:    2;  —  5  =  2.     Is  2;  a  known 

or  an  unknown  quantity?     What  is  5?     What 
is  2? 

3.  Make  an  equation  to  show  that  4  added  to  a 

certain  quantity  equals  12. 

4.  Make  an  equation  to  show  that  8  subtracted  from 

a  certain  quantity  equals  2. 

5.  If  x  +  2  =  5,  5  is  how  much  more  than  xl     Does 

this    equation    express    the    same    condition: 
X  =  5  —  2?     What  is  the  value  of  a:? 

6.  If  you  subtract  4  from  both  members  of  this  equa- 

tion:   2/  +  4  =  9,  how  will  the  equation  read? 
What  is  the  value  of  2/? 

7.  Solve  this:  ;2  +  3  =  6. 

8.  Solve  this:  a  +  9  =  10. 

9.  Solve  this:  &  +  7  =  11. 

10.  Solve  this:  c  +  4  =  14. 

11.  Make  an  equation  to  show  that  all  the  chairs  in 

a  room  and  3  chairs  more  make  a  total  of  9  chairs. 

12.  Make  an  equation  to  show  that  all  the  water  in 

a  tank  excepting  10  gallons  equals  90  gallons. 

13.  Make  an  equation  to  show  that  all  the  oats  in 

a  bin  excepting  7  bushels  equals  18  bushels. 

(VIII-46) 


EQUATIONS 

Lesson  14 
Solving  Equations  (Adding  and  Subtracting) 

EXAMPLE: 

X  +  3  =  12 

-          3         3 

X           =9 

EXAMPLE: 

X  +  3  =  6 

+          33 

X  +  6  =  9 

Since  the  two  members  of  an  equation  are  equal  in 
value,  it  is  easily  understood  that  if  we  add  the  same 
number  to  both  members,  or  if  we  subtract  the  same 
number  from  both  members,  the  equality  will  not  be 
disturbed. 


EXAMPLE: 

X  +  3  =  12 
same  as  x  =12  —  3; 

because  x  =9 


EXAMPLE: 

X  +  3  =  6 
same  as  x  =6  —  3; 

because  x  =3 


The  same  result  is  obtained  by  transposing  a  quantity 
from  one  member  to  the  other,  at  the  same  time  chang- 
ing the  sign  from  +  to  —  or  from  —  to  +  as  the  case 
may  be. 

In  solving  equations,  first  transpose  all  the  known 
quantities  to  the  second  member,  changing  signs  from 
-f  to  —  or  from  —  to  +  as  is  necessary. 

In  all  equation  work,  remember  that  unlike  numbers 
must  be  kept  separate: 

3x  -\-  2x  =  5x;  Sx  —  2x  =  x; 

3a:  +  3  =  3a;  +  3;  3x  -  2  =  3x  -  2; 

3x  -{- 2x -\-  5x  =  lOx;  3a:  +  2a;  +  2y  =  5a-  +  2y; 

3x  -  X  -  2y  =  2x  -2y;  4:Z  +  5z  =  9z. 

(VIII-47) 


ARITHMETIC 


Exercise  29 — Oral. 

First  sa}^  how  you  will  transpose  the  known  quantities 
to  the  second  member,  then  solve: 


1. 

X   +   4:    = 

8 

;    11. 

14  = 

c  +  5 

2. 

2/-3  = 

7 

;    12. 

18  = 

y-   2 

3. 

2  +  8  = 

12 

;    13. 

7  = 

x+  1 

4. 

a  +  5  = 

10 

;    14. 

11  = 

y  -    5 

5. 

h  -4  = 

3 

;    15. 

12  = 

z-\-10 

6. 

5  +  c  = 

11 

16. 

18  = 

10+    h 

7. 

8  +  x  = 

16 

17. 

14  = 

17  -  z 

8. 

12  +  2/  = 

15 

18. 

13  = 

1  +  X 

9. 

10  +  ^  = 

11; 

19. 

11  = 

3  +  ^ 

10. 

8  -  X  = 

6; 

20. 

12  = 

15-  y. 

21.  What  number  increased  by  8  equals  12? 

22.  What  number  decreased  by  9  equals  6? 

23.  What  number  is  5  more  than  10? 

24.  What  number  is  12  less  than  20? 

25.  If  X  +  20  =  35,  what  is  the  value  of  x? 


Solve : 

26.  3a;  +  4a;  =  ? 

27.  Qy  -  4y  =  ? 

28.  72  +  3  =  ? 

29.  4a:  -  2  =  ? 

30.  5x  +  2y  =  ? 

31.  2^  +  4^  +  2;  =  ? 

32.  a  +  2a  +  3a  =  ? 

33.  z  +  4:Z  -  2z  =  ? 

34.  y  -}-  5y  -  y  =  ? 

35.  a:  +  a:  +  3x  =  ? 

36.  3y  +  2y  =  10; 


37.  5a:  —  a:  =  12; 

38.  2z  +  4z  +  5  =  17; 

39.  4y  -  y  -  3  =  9; 

40.  3a:  =  10  +  x; 

41.  Sz  =  20  -  2z; 

42.  5y  +  2y  =  24  -  y; 

43.  7z  -  3z  =  9  +  z; 

44.  9a:  +  5a:  -  24  +  2a:; 

45.  4y-\-3y  +  4  =  20-  y; 

46.  3b  +  4b  =  24  -b; 

47.  5c  -  3c  =  10  +  c. 


(VIII-48) 


EQUATIONS 

Lesson  15 
Solving  Equations  (Multiplying  and  Dividing) 


EXAMPLE: 

EXAMPLE: 

a;  =  3 

la:  =    4 

2)2x  =  6 

X              3 

a:  =  12 

Since  both  members  of  an  equation  are  equal  in  value, 
if  we  multiply  both  members  by  the  same  number,  or 
divide  both  members  by  the  same  number,  the  equality 
will  not  be  disturbed. 

Since  x  -{-  x  +  x  =  Sx,  then  3  X  ^  must  also  equal 
3x;  therefore,  an  expression  such  as  3a;  means  3  X  x, 
but  expressions  of  this  kind  are  always  wTitten  without 
the  sign  of  multiplication.  4y  means  4  X  y;  Sz  means 
8X2;  etc.  y  -T-  3  may  be  written  as  iy^  or  as  I; 
z  -i-  4:  =  iz,  or  I;  etc. 


EXAMPLE: 


3 


A;  L.  C.  D.  (6);?^ 

2  6 

2x  =  'S:  X  =  U 


When  there  are  denominators  in  the  equation,  both 
members  must  be  reduced  to  the  L.  C.  D.  and  then 
the  denominators  can  be  dropped  without  disturbing 
the  equality.    This  step  is  called  ''clearing  of  fractions.'' 


EXAMPLE: 

2a:  =  6 
same  as    x  =  6  -^  2; 
because    a:  =  3 


EXAMPLE: 


ix  =  4 


same  as    x  =  4  X  3; 
because    x  =  12 


(VIII-49) 


ARITHMETIC 

Here  again,  we  can  obtain  the  same  result  by  trans- 
posing a  quantity  from  one  member  to  the  other,  at 
the  same  time  changing  the  sign  from  -^  to  X  or  from 

X  to  -T-  as  the  case  may  be. 

Exercise  30 — Oral. 

First  say  how  you  will  solve  each  equation,  then  give 
the  value  of  x,  y,  or  z: 


1. 

4x    =  12; 

9. 

12 

=    ix; 

2. 

5y    =  10; 

10. 

15 

=  liy; 

3. 

2z    =  20; 

11. 

1 

X  . 

4. 

24       =     6a:; 

12. 

1 

=       12; 

5. 

32      =    8^; 

13. 

2 

6. 

i2/  =    4; 

14. 

X 

T 

1  . 

7. 

ix  =     5; 

15. 

X 

_           2  . 
—          7, 

8. 

U  =     6; 

16. 

y 
7 

=  5; 

Solve : 

17. 

3X2/  =  ? 

25. 

x 

■^  3  =  ? 

18. 

4  X  ^  =  ? 

26. 

y 

-^  8  =  ? 

19. 

12  X  a  =  ? 

27. 

^  =  ^- 

20. 

8X6  =  ? 

28. 

ix  =  ? 

21. 

fc  X  8  =  ? 

29. 

1=  ?-^  ? 

22. 

c  X  3  =  ? 

30. 

iz  =  ?  ^? 

23. 

re  X  5  =  ? 

31. 

a 

^  6  =  ? 

24. 

100  X  c  =  ? 

32. 

12-  —  : 

Exercise  31 — Oral  and  Written. 

A.  State  the  equation  that  you  will  use  in  solving 
each  of  these  problems: 

1.  The  distance  between  two  trees  is  5  times  the  dis- 
tance between  one  of  the  trees  and  a  shrub.     If 

(VIII-50) 


EQUATIONS 

the  distance  between  the  tree  and  the  shrub  is 
10  yd.,  how  far  apart  are  the  two  trees? 

(Let  X  represent  the  distance  between  the  two 
trees;  therefore,  Ix  =  10  yd.,  and  a;  =  50  yd., 
Ans.     Proof:  i  of  50  yd.  =  10  yd.) 

2.  I  have  two  books,  one  with  a  red  cover  and  one 

with  a  blue  cover.  The  red  book  weighs  5  oz. 
less  than  6  times  as  much  as  the  blue  book  and 
the  weight  of  the  blue  book  is  3  oz.  What  is 
the  weight  of  the  red  book? 

3.  The  minute  hand  on  a  watch  moves  12  times  as 

fast  as  the  hour  hand.  The  two  hands  point  in 
exactly  the  same  direction  at  12  o'clock.  What 
time  will  it  be  when  they  next  point  in  exactly 
the  same  direction? 

(Let  X  represent  the  distance  the  hour  hand 
must  move.) 

4.  The  height  of  a  certain  oak  tree  is  8  times  the 

height  of  John,  and  John  is  \  as  tall  as  a  young 
poplar  tree.  If  the  height  of  the  poplar  tree  is 
12  ft.,  what  is  the  height  of  the  oak  tree? 

5.  One-half  of  a  certain  number  is  3  more  than  one- 

third  of  the  number.     What  is  the  number? 

6.  The  sum  of  two  numbers  is  22,  and  one  of  the 

numbers  is  2  less  than  the  other;  what  are  the 
tw^o  numbers? 

7.  The  width  of  Park  Avenue  is  30  ft.  from  curbing 

to  curbing.     The  combined  width  of  both  side- 
walks is  7  ft.  less  than  one-half  the  entire  width 
of  the  street  including  the  sidewalks.     What  is 
the  width  of  each  sidewalk? 
CVIII-51) 


ARITHMETIC 

8.  The  perimeter  of  a  rug  is  54  ft.;  if  the  difference 

between  the  length  and  the  width  is  3  ft.,  what 
are  the  dimensions  of  the  rug? 

9.  The  perimeter  of  a  building  lot  is  450  ft.;   if  the 

combined  length  of  the  two  long  sides  is  50  ft. 
greater  than  the  combined  length  of  the  two 
short  sides,  what  are  the  dimensions  of  the  lot? 
10.  The  height  of  a  room  is  3  ft.  less  than  the  length 
but  2  ft.  more  than  the  width;  if  the  combined 
length,  width  and  height  is  37  ft.,  what  are  the 
dimensions  of  the  room? 
B.  Now  solve  the  problems;  also  prove  each  answer. 


(Vin-52) 


I 


MENSURATION 

Lesson  16 

Right  Triangles 

Read  carefully. 

A  right  triangle  is  a  triangle  having  one  right  angle. 
The  two  other  angles  of  a  right  triangle  are  acute  angles. 

The  two  sides  which  form  the  right 
angle  are  called  the ''legs"  of  the  tri- 
angle, and  the  remaining  side  is  called 
the  ^^ hypotenuse";  thus,  in  the  right 
triangle  ABC  shown  in  Figure  1,  the 
sides  AB  and  AC  are  the  legs  and  EC 
is  the  hypotenuse.  "^    (Figure  i) 

Over  two  thousand  years  ago,  a  Greek  mathematician 
named  Pythagoras  discovered  the  fact  that  when 
squares  are  drawn  on  all  three  sides  of  a  right  triangle, 
the  area  of  the  square  on  the  hypotenuse  is  equal  to 
the  sum  of  the  areas  of  the  squares  on  the  two  other 
sides.     This  rule  is  known  as  the  Pythagorean  theorem. 

To  prove  that  this  is  so,  we  need  only  to  make  three 
cardboard  squares  to  correspond  with  the  three  sides 
of  a  right  triangle  as  shown  in  Figure  2.  Taking  the 
square  which  is  neither  the  largest  nor  the  smallest, 
that  is  Square  A,  find  the  point  where  its  diagonals  cross, 
that  being  the  exact  center  of  the  square.  Now  divide 
and  cut  this  square  into  four  equal  parts  by  drawing 
two  lines  through  the  center  of  the  square,  the  first  line 
running  parallel  to  the  hypotenuse  of  the  given  triangle, 

(VIII-53) 


ARITHxMETIC 


and  the  second  line  crossing  the  first  Une  at  right  angles, 
as  shown  in  Figure  3.  Now  lay  Square  B  and  the  four 
parts  of  Square  A  on  vSquare  C  as  indicated  in  Figure  3. 


(Figure  2)  (Figure  3) 

It  naturally  follows,  that  since  the  square  on  the 
hypotenuse  (C  =  25  sq.  in.)  is  equal  to  the  sum  of  the 
squares  on  the  two  other  sides  (A  =  16  sq.  in.  +  J5  =  9 
sq.  in.)  then  C  —  B  =  A,  and  C  —  A  =  B;  in  other 
words,  the  square  on  the  hypotenuse  minus  the  square 
on  either  leg  equals  the  square  on  the  other  leg. 

Since  the  length  of  one  side  of  a  square  is  equal  to 
the  square  root  of  the  area,  we  can  find  the  length  of 
any  side  of  a  triangle  when  we  know  the  area  of  the 
square  on  that  side  by  extracting  the  square  root  of 
such  area.  Thus,  in  Figure  2,  if  we  know  that  the 
area  of  A  is  16  sq.  in.,  we  know  that  the  length 
of  that  side  is  Vl6  in.,  or  4  in.;  in  like  manner, 
the  area  of  jB  is  9  sq.  in.,  therefore,  the  length  of  that  side 
is  \/9  in.,  or  3  in.;  and  the  area  of  C  is  25  sq.  in., 
therefore,  the  hypotenuse  is  \/25  in.,  or  5  in. 

Count  the  blocks  in  the  three  squares  shown  in 
Figure  2  above?     Is  it  true? 

(VIII-54) 


v^ 


MENSURATION 

Make  one  of  your  own  and  use  6"  and  8"  as  the  legs 
to  find  the  hypotenuse.  Prove  by  blocking  it  as  he  did. 
Now  prove  by  cutting  it  and  fitting  the  parts. 

The  sign  A  is  often  used  to  indicate  a  triangle;  A 
for  triangles. 

Exercise  32 — Oral. 

The  children  of  one  row  are  to  ask  the  following 
questions  of  the  others: 

1.  Referring  to  the  triangle  shown  in  Figure  2,  state 

the  length  of  each  of  the  two  legs.  Of  the 
hypotenuse. 

2.  WTiat  kind  of  a  triangle  is  this?     Why? 

3.  What  is  the  ratio  of  a  square  constructed  on  the 

hj^potenuse  of  a  right  triangle  to  the  sum  of 
squares  constructed  on  the  two  legs? 

4.  State  the  rule  covering  the  theorem  discovered  by 

Pythagoras  regarding  squares  constructed  on  the 
sides  of  right  triangles. 

5.  Again  referring  to  Figure  2,  how  can  we  find  the 

area  of  Square  C  when  we  know  the  areas  of  A 
and  B?  How  can  we  find  A  when  we  know 
B  and  C?  How  can  we  find  B  when  we  know^ 
A  and  C? 

6.  How  can  we  find  the  square  on  either  leg  of  a 

right  triangle  when  the  squares  on  the  hypot- 
enuse and  on  the  other  leg  are  known? 

7.  If  we  know  the  length  of  any  side  of  a  A,  how 

do  we  find  the  square  on  that  side? 

8.  If  we  know  the  square  on  any  side  of  a  A,  how 

do  we  find  the  length  of  that  side? 

(VIII-55) 


ARITHMETIC 

9.  If  the  square  on  the  hypotenuse  is  64  sq.  m.,  what 

is  the  length  of  the  hypotenuse? 
10.  If  the  squares  on  the  two  legs  of  a  right  triangle 
are  respectively  36  sq.  in.  and  64  sq.  in.,  what  is 
the  area  of  the  square  on  the  hypotenuse?  What 
is  the  length  of  the  hypotenuse?  What  is  the 
length  of  each  of  the  two  legs? 

Exercise  33 — Written. 

1.  The  legs  of  a  right  triangle  are  respectively  15  ft. 

and  20  ft. ;  what  is  the  length  of  the  hypotenuse? 

2.  The  hypotenuse  of  a  right  A  is  95  ft.  and  one  of 

the  legs  is  57  ft.;  what  is  the  length  of  the 
other  leg? 

3.  The  hypotenuse  of  a  right  triangle  is  22 1  in.  and 

one  of  the  legs  is  13 J  in.;  what  is  the  length  of 
the  other  leg? 

4.  The  legs  of  a  right  A  are  respectively  6  ft.  and 

8  ft.  long;  what  is  the  length  of  the  hypotenuse? 
(Make  a  chalk  mark  on  the  floor  8  feet  from 
a  wall,  and  make  one  on  the  wall  6  feet  from  the 
floor;  prove,  by  using  a  string  of  the  length 
corresponding  to  the  hypotenuse  of  this  triangle, 
that  the  wall  is  perpendicular.) 

5.  Using  the  principle  outlined  in  Question  4,  open 

a  door  so  that  it  forms  an  exact  right  angle 
with  the  wall. 

6.  The  legs  of  a  compass  are  5  in.  long;  what  is  the 

distance  from  the  point  of  one  leg  to  the  point 
of  the  other  leg  when  the  compass  is  opened  to  a 
right  angle? 

(VIII-56) 


MENSURATION 

7.  A  smoke-stack  40  ft.  high  is  to  be  held  rigid  by 

4  wires  running  from  the  top  of  the  stack  to 
points  30  ft.  from  the  bottom  of  the  stack; 
allowing  10  ft.  for  fastening  each  of  these  4 
wires,  what  is  the  total  length  of  the  wire  required 
to  anchor  this  stack? 

8.  A  baseball  diamond  is  90  ft.  square;  how  far  is  it 

from  first  base  to  third  base? 

9.  What  is  the  length  of  the  diagonal  of  a  20-ft. 

square? 

10.  The  area  of  a  square  lot  is  15,625  sq.  ft. ;  what  is 

the  length  of  its  diagonal? 

11.  Find  the  longest  line  in  a  rectangle  20  yd.  long 

and  12  i  yd.  wide. 

12.  Find  the  diagonal  of  the  ceihng  of  a  room  40' 

long  and  35'  wide. 

13.  Find  the  length  of  the  line  from  the  upper  corner 
,  of  a  room  to  the  lower  corner  diagonally  oppo- 
site;   the  room  measurements  are:    length,  45 
ft.;  height,  15  ft.;  width,  32  ft. 

(Suggestion:    Get  the  diagonal  of  the  ceiling 
first.) 

14.  Find  the  volume  of  a  cube  whose  entire  surface  is 

486  sq.  in. 

Lesson  17 

Isosceles  and  Equilateral  Triangles 

A  triangle  having  three  equal  sides  is  an  '^  equilat- 
eral'^ triangle. 

A  triangle  having  two  equal  sides  is  an  'isosceles" 
triangle;  therefore,  every  equilateral  triangle  is  also 
isosceles. 

(VIII-57) 


ARITHMETIC 


A  right  triangle  may  be  isosceles,  but  can  it  ever  be 
equilateral?     Why  not? 


VERTEX 


VERTEX 


ALTITUDE 


-ALTITUDE 


Equilateral 
Triangle 


Isosceles 
Triangle 


Read  carefully  so  you  can  question  well. 

A  straight  line  drawn  from  any  vertex  of  an  equi- 
lateral triangle  to  the  middle  of  the  opposite  side  is 
perpendicular  to  that  side,  and  divides  the  side  as  well 
as  the  triangle  into  two  equal  parts. 

A  straight  line  drawn  from  the  vertex  where  the  two 
equal  sides  of  an  isosceles  triangle  meet  to  the  middle 
of  the  base  is  perpendicular  to  the  base,  and  divides 
the  base  as  well  as  the  triangle  into  two  equal  parts. 


APOTHEM 


APOTHEM 

A  Regular  Pentagon 


A  Regular  Hexagon 

A  regular  hexagon  is  composed  of  six  equilateral 
triangles. 

A  regular  pentagon  is  composed  of  five  isosceles 
triangles. 

(VIII-58) 


MENSURATION 

I  A  perpendicular  line  drawn  from  the  center  of  one 
of  the  sides  to  the  center  of  a  many-sided  figure  is 
called  an  ''apothem";  it  corresponds  to  the  altitude 
of  a  triangle. 

Exercise  34 — Oral. 

1.  Draw  an  equilateral  A  on  the  board. 

2.  Draw  an  isosceles  A. 

3.  Draw  a  right  triangle  which  is  also  an  isosceles  A. 

4.  Show^  that  a  right  triangle  can  also  be  an  isosceles 

A. 

5.  Draw  an  altitude  in  an  isosceles  A.     Tell  what 

it  does  to  the  base.     To  the  A. 

6.  Has  every  triangle  an  altitude?     Tell  about  it. 

7.  A  regular  hexagon  has  how  many  sides? 

8.  Of   how   many   triangles   is   a   regular   hexagon 

formed?     Of  what  kind  of  triangles  is  it  formed? 

9.  A  regular  pentagon  has  how  many  sides?     How 

many  A?    What  kind  of  A? 

10.  Of  how  many  triangles  is  a  regular  octagon  formed? 

11.  What  is  an  apothem  of  a  hexagon? 

12.  What  is  the  altitude  of  a  triangle? 

13.  An   equilateral   triangle    has    how    many   equal 

sides?     How  many  equal  angles? 

14.  An  isosceles  triangle  has  how  many  equal  sides? 

How  many  equal  angles? 

Exercise  35 — Written. 

1.  The  base  of  an  isosceles  triangle  is  6  ft.  and  the 
altitude  is  4  ft.;  what  is  the  length  of  each  of 
the  two  equal  sides? 

(Vin-59) 


ARITHMETIC 

2.  The  base  of  an  isosceles  triangle  is  36  ft.  and  each 

of  the  two  equal  sides  is  30  ft.  in  length;  what 
is  the  altitude? 

3.  What  is  the  area  of  the  A  referred  to  in  Question  1? 

4.  What  is  the  area  of  the  A  referred  to  in  Question  2? 

5.  What  is  the  altitude  of  an  equilateral  triangle 

having  sides  10"  long?  If  you  find  the  altitude 
is  8.66  +  in.  when  the  sides  are  10  in.  long, 
what  would  be  the  altitude  for  each  V  of  side? 

6.  Knowing    that    the    altitude    of    an    equilateral 

triangle  equals  .866  of  the  side,  find  the  altitude 
of  an  equilateral  triangle  having  25-inch  sides. 

7.  What  is  the  area  of  an  equilateral  triangle  having 

sides  20''  in  length? 

8.  What  is  the  area  of  a  regular  hexagon  having  sides 

r  long? 

9.  What  is  the  area  of  the  largest  regular  hexagon 

which  can  be  constructed  in  a  circle  having  a 
a  6"  radius? 
10.  A  regular  pentagon  whose  sides  are  12"  long  is 
constructed  in  a  circle  having  a  9"  radius;  what 
is  the  area  of  this  pentagon? 

Lesson  18 

Similar  Triangles 

Triangles  that  have  the  same  shape,  though  they  be 
different  in  size,  are  called  '^similar  triangles." 

Naturally,  the  corresponding  angles  of  similar  tri- 
angles are  equal,  and  the  corresponding  sides  of  similar 
triangles  are  proportional,  otherwise  the  triangles  could 
not  have  the  same  shape. 

(VIII-60) 


MENSURATION 


(Figure  2) 


(Figure  3) 


That  Triangle  xyz  (Figure  1)  is  similar  to  Triangle 
XYZ  (Figure  2)  is  proven  by  the  fact  that  each  of  the 
angles^of  Triangle  xyz  is  equal  to  the  corresponding 
angle  of  Triangle  XYZ  as  is  shown  by  Figure  3  where 
Triangle  xyz  is  placed  on  Triangle  XYZ  in  three 
different  positions.  That  these  two  triangles  are  similar 
is  further  proven  by  the  fact  that  each  of  the  sides  of 
Triangle  XYZ  is  three  times  as  long  as  the  correspond- 
ing side  of  Triangle  xyz. 


(Figure  4) 


(Figure  5) 


c 
(Figure  6) 


Test :  See  if  the  angles  of  Figure  4  coincide  with  the 
corresponding  angles  of  Figure  5.  Cut  your  own  A  and 
fit  them  to  coincide  or,  as  we  say,  ^'superimpose"  one 
on  the  other.  Try  placing  your  smaller  triangle  in 
three  different  positions  as  shown  in  Figure  3. 

Knowing  that  corresponding  sides  of  similar  triangles 
are  proportional,  it  is  quite  evident  that  if  we  know  the 
length  of  any  two  sides  of  a  given  triangle  and  the 
length  of  one  corresponding  side  of  a  similar  triangle, 

(VIII-61) 


ARITHMETIC 

the  length  of  the  other  corresponding  side  may  be  found 
by  proportion;  thus: 


EXAMPLE:     Find  the  length  of  AE,  if— 

Triangle  ABC,  AB  =  5";  AC  =  4'; 
Triangle  ADE,  AD  =  1Y\  AE  =  ?; 

7|:  5  =  ?:  4;  or^  =  1; 
5        4 

Using  L.  C.  D.  (20),  |^  =  ^; 

'^  =  J?;  therefore  (.'.)  AE  =  6^  Ans. 
20        4' 


In  this  example  the  length  of  AE  is  an  unknown 
quantity  which  we  are  required  to  find.  Let  us  call 
this  unknown  quantity  x  and  see  if  we  can  work  the 
example  a  little  differently : 


EXAMPLE:     Find  the  length  of  x,  if— 

Triangle  ABC,  AB  =  5";  AC  =  4"; 
Triangle  ADE,  AD  =  7V;  AE  =  x; 

7|:5  =  x:4;    ot  1i  =  - 
5        4; 

Reducing  to  L.  C.  D.  (20)  we  have  ^  =  -|^;  f  =  ^' 

Tf  2r  =  _  ,  then  30  =  5a:  or  5  times  the  unknown  quantity; 
^^  20       20 

If  30  is   5  times   the   unknown   quantity,   then   the   unknown 

quantity  is  I  of  30,  or  6;  .*.  AE  =  6",  Ans, 


Another  example  based  on  the  same  two  triangles 
(Figures  4  and  5)  follows.  In  this  example  we  are 
required  to  find  the  length  of  DE,  while  in  the  pre- 
vious example,  we  found  the  length  of  AE. 

(VIII-62) 


MENSURATION 


I 


EXAMPLE: 

Find  the  length  of 

X,  if- 

Triangle  ABC, 

AC 

=  4" 

;  BC 

=  3"; 

Triangle  ADE, 

AE 

=  6 

']  DE 

=  x; 

6:4  =  x:3; 

or   5=f;L.C 
4       3 

.  D. 

(12) 

.  18 
'  12 

4x. 
"  12' 

After  bringir 

ig  to  L.   C.  D. 

we 

can 

drop 

the  denominator  or 

"clear  of  fractions,"  therefore, 

18  = 

4x; 

If  18  =  4x,  X 

=  i  of  18  or  4A 

DE 

=  4-', 

Ans. 

Triangles  may  occupy  various  positions  and  still  be 
similar  and,  therefore,  proportional.  Thus,  if  Triangle 
ABC  were  inverted  and  attached  to  Triangle  ADE,  the 
figure  would  appear  as  follows: 


(Figure  7) 

Are  the  same  corresponding  sides  here? 
Are  the  same  corresponding  angles  here?     Are  they 
equal? 

Does  this  position  affect  their  similarity? 
Are  the  figures  similar? 

(VnT-63) 


ARITHMETIC 


Many  practical  measurements  can  be  made  by  the 
use  of  triangles  as  is  shown  by  the  following: 

H-rE.O    POINT^ 
M    B   AND      E.y 


(Figure  8) 

Let  US  assume  that  we  wish  to  measure  the  distance 
across  a  lake  DE.  We  lay  out  the  line  CE  with  one 
end  at  E,  and  make  it  long  enough  so  that  we  may 
construct  similar  triangles  thereon;  we  then  lay  out 
the  line  CB  perpendicular  to  CE  and  parallel  to  DE, 
By  standing  at  B  and  looking  at  the  point  D  which  we 
recognize  by  some  peculiarity  of  land  or  plant  forma- 
tion, we  locate  the  point  A  where  BD  crosses  CE. 

We  now  measure  and  find  CA  to  be  20  yd.,  A £^  to  be 
50  yd.,  and  CB  to  be  30  yd. ;  this  enables  us  to  find  the 
length  of  DE  by  the  following  example  in  proportion: 


EXAMPLE: 

Find  the  length  of  x, 

if— 

Triangle  ABC,  AC  = 

=  20yd.; 

BC  = 

30  yd. 

> 

Triangle  ADE,  AE  = 

=  50  yd.; 

DE  = 

=  x; 

Then  50:20 

=  x:30;    or  ^  =  ^; 
20       30' 

L.  C.  D 

(60); 

150  _ 
60 

2x 

60 ; 

Clearing  of  fractions,  150  =  2a:; 

Dividing  by  ' 

I,  75  =  x;  .'.  DE  =  75  yd.,  Am 

3. 

(VIII-64) 


MENSURATION 

Exercise  36 — Oral  and  Written.  ' 

Some  boy  may  draw  two  similar  triangles  on  the 
board.  Members  of  2  rows  may  do  the  questioning. 
Ask  these  questions  and  others: 

1.  Triangles  of  the  same  shape  have  what  name? 

2.  Must  similar  triangles  be  of  the  same  size? 

3.  What   are   corresponding   sides? 

4.  If  one  side  of  a  given  triangle  is  exactly  twice  as 

long  as  the  corresponding  side  of  a  similar  tri- 
angle, how  will  the  two  other  sides  of  the  given 
triangle  compare  with  the  two  other  correspond- 
ing sides  of  the  similar  triangle? 

5.  Draw  two   similar  triangles,  one  having  sides  } 

longer  than  the  corresponding  sides  of  the  other. 

6.  Draw  two  similar  A,  one  of  which  has  sides  75% 

longer  than  the  corresponding  sides  of  the  other. 

7.  Must  similar  triangles  occupy  similar  positions? 

8.  Draw  two  similar  triangles,  one  of  which  has  sides 

bearing  the  ratio  of  1  to  2  to  the  corresponding 
sides  of  the  other,  and  let  one  triangle  occupy  an 
inverted  position  in  relation  to  the  other. 

9.  In  Figure  8,  what  would  be  the  length  of  DE  if 

AE  were  40  yd.?     60  yd.?     80  yd.? 

10.  In  Figure  8,  what  would  be  the  length  of  DE  if 

CB  were  20  yd.?    40  yd.?     60  yd.? 

11.  If  two  of  the  angles  of  a  triangle  are  equal  to  the 

two  corresponding  angles  of  another  triangle, 
could  the  third  corresponding  angles  be  unequal? 
Draw  several  similar  triangles  and  explain  fully. 

12.  If  x  =  10  yd.,  what  is  the  value  of  2a:?     Of  5x? 

Of  lOx?     Of  ix?     Of  ix?     Of  tVx? 

(VIII-65) 


ARITHMETIC 


13.  If  X  represents  an  unknown  quantity,  what  does 
Zx  represent?     8a: ?     \x1 
If  24  =  4a:,  what  is  the  value  of  x1    Of  5a:?    Of  \x1 
If  To  =  A,  what  is  the  value  of  a:? 
If  ^  =  A,  what  is  the  value  of  a:? 


14. 
15. 
16. 
17. 


If  I  =  I,  what  must  be  done  before  we  can  find 


18. 
19. 


the  value  of  a:?      Reduce   I  = 
Now  find  the  value  of  x. 

If  I  =  I,  what  is  the  value  of  a:? 

Read  and  solve:  15  :  3  =  a:  :  3. 


I  to  L.  C.  D. 


10  :  2  =  a:  :  4. 


20.  What  is  the  meaning  of  proportional? 
Exercise  37 — Written. 


1.     Two  boys  had  a  friendly  argument   regarding 
the  height  of  the  top  of  the  flag-staff  shown  in 

(VIII-6G) 


MENSURATION 

the  picture.  Fred  said  he  thought  the  height 
to  be  about  40  ft.  from  the  ground,  and  John 
thought  it  to  be  about  70  ft.,  so  they  decided 
to  ascertain  its  height  by  the  use  of  similar 
triangles.  John  took  a  10-ft.  stick  and  stood  54 
ft.  from  a  point  directly  under  the  flag-staff;  Fred 
had  to  stand  6  ft.  farther  from  the  staff  to  be  able 
to  sight  the  top  of  the  staff  over  the  top  of  the 
10-ft.  stick,  and  Fred's  eyes  were  exactly  5  ft. 
from  the  ground;  what  was  the  height  of  the 
top  of  the  flag-staff  from  the  ground? 

2.  If  Fred  had  to  stand  5  ft.  instead  of  6  ft.  farther 

from  the  staff  than  John  to  be  able  to  sight  the 
top  of  the  staff  over  the  10-ft.  pole,  what  would 
be  the  height  of  the  top  of  the  staff  from  the 
ground? 

3.  Whsit  is  the  height  of  a  tree  w^hich  casts  a  shadow 

150  ft.  long  when  a  boy  4  ft.  2  in.  tall  casts  a 
shadow  4  ft.  long? 

4.  A  certain  mountain  has  its  highest  peak  exactly 

over  the  center  of  the  base,  and  the  base  is  a 
quarter  of  a  mile  wide;  how  high  is  the  highest 
peak  if  it  can  be  seen  over  the  top  of  a  25-ft. 
tree  which  stands  1,500  yards  from  the  base  of 
the  mountain,  when  the  observer  whose  eyes  are 
5  ft.  from  the  ground  stands  40  yards  beyond  the 
tree? 

5.  If  a  card  7^  in.  wide  held  2  ft.  from  one  of 

your  eyes  enables  you  to  sight  the  width  of  a 
door  which  is  6  ft.  beyond  the  card,  how  wide 
is  the  door?     (Close  one  eye  while  trying  this.) 

(VIII-67) 


ARITHMETIC 

6.  If  a  card  4"  wide  held  2'  6''  from  the  eye  com- 
pletely hides  the  width  of  a  picture  3'  wide  from 
sight,  how  far  from  the  eye  is  the  picture? 


Lesson  19 
Table  of  Angular  Measure 

60  seconds  (") =  1  minute  (') 

60  minutes =  1  degree  (°) 

90  degrees =1  right  angle  ( L ) 


VE.RTEX 

f\     PROTRACTOR 


The  instrument  commonly  used  for  measuring  angles 
and  arcs  is  called  a  ''protractor"  (see  illustration). 
To  measure  an  angle  or  arc  place  the  vertex  of  the 
angle  at  the  center  of  the  protractor  with  one  side  of 
the  angle  running  along  the  0-line  of  the  protractor; 
the  reading  where  the  other  side  of  the  angle  falls  is 
the  number  of  degrees  in  the  angle  or  arc.  The  angle 
AVX  =  40°. 

The  sign  for  an  angle  is  Z  ;  for  several  angles  Z^. 
The  sign  for  a  right  angle  is  L  ;   for  several  right 
angles  L! . 

(VIII-68) 


jMENSURATION 


Exercise  38 — Oral  and  Written. 

(Make  a  protractor  or  buy  one.) 

1.  Draw  a  circle.     Locate  C;   D;   R;   show  an  arc; 

a  quadrant;  a  sector. 

2.  How  many  °  are  there  in  a  circle?      (Use  pro- 

tractor.) 

3.  How  many  °  are  there  in  a  right  angle? 

4.  How  many  right  angles  are  there  in  a  circle? 

5.  Are  the  sides  equal  in  these  three  li  ?     Prove. 

6.  Measure  these   Li. 


^ 


bi 


7.  Did  the  different  lengths  of  sides  affect  the  li  ? 

8.  Draw  a  quadrant.     Take  |  of  it  with  protractor. 

9.  How  many  °  are  there  in  ^  of  J  of  a  circle? 

10.  J  of  J  of  a  circle  is  what  part  of  a  circle? 

11.  A  What     is    the    point    called 

where  the  two  sides  of  an 
Z  meet?  An  angle  is  read 
with  this  point  in  the 
center:  as  lABC  or 
LCBA. 

12.  Measure  these  angles 

with       your      pro- 
tractor : 

13.  What  is  the  sum  of  these  two  angles? 

14.  What  is  the  difference  between  these  two  angles? 

CV^III-69) 


ARITHMETIC 


Exercise  39 — Written. 

1.  Draw  a  triangle. 

2.  Letter  the  Z'  A,  B,  C. 

3.  Measure  each  carefully.    Put  your  results  on  your 

drawing. 

4.  Find  the  sum  of  the  3  Z^. 

5.  Cut  out  the  3  Z^  carefully  and  place  all  vertices 

(plural  of  vertex)  at  one  point  on  a  straight  line. 

6.  Does  this  prove  your  answer  to  Question  4? 

7.  This  makes  how  many  right  angles? 

8.  When  it  makes  2  right  angles  call  it  a  straight 

angle.     (180°) 

Lesson  20 

Measuring  the  Length  of  Arcs  and  the  Area  of  Sectors 

of  Circles 

Read  carefully. 
You  have  learned  that 
the  circumference  of  a 
circle  is  equal  to  ^'pi'' 
(tt)  times  the  diameter, 
and  you  also  have 
learned  that  there  are 
360°  in  every  circumfer- 
ence; hence,  to  find  the 
length  of  any  arc,  we 
first  find  the  length  of 
the  entire  circumference 
of  which  the  arc  is  a  part,  and  then  we  find  the  required 
number  of  360ths  of  such  circumference  corresponding 
to  the  number  of  degrees  in  the  arc. 

(VIII-70) 


CSCALE.  'A) 


MENSURATION 

In  the  figure  here  shown,  the  circle  has  a  34"  radius 
or  a  1"  diameter,  and  the  circumference,  therefore, 
equals  7"  X  3.1416  =  22''  (nearly),  and  45°  is  ^  or 
i  of  360°;   therefore,  the  length  of  this  arc  is  i  of  22" 

ri^tf 

—    -^4     • 

To  find  the  area  of  a  sector  of  a  circle,  we  must  first 
find  the  area  of  the  entire  circle  and  then  take  the 
required  number  of  360ths  of  such  area  corresponding 
to  the  number  of  degrees  in  the  sector. 

In  the  figure  here  shown,  we  find  the  circumference 
to  be  22'',  and  the  area  is  found  by  multiplying  the 
circumference  by  4  the  radius;  therefore,  22"  X  If 
=  38i  sq.  in.  area,  and  this  sector  contains  45°  or  3^ 
of  the  entire  area;  -Ho  =  i;  i  of  38i  sq.  in.  =  4.8125 
sq.  in.,  area  of  the  sector. 

Another  way  of  finding  the  area  of  a  circle,  and  one 
which  is  more  generally  used  when  the  diameter  or 
radius  is  known,  is  to  multiply  ^'pi"  by  the  square  of 
the  radius;  thus,  3.1416  X  (34)^  =  38.5  sq.  in.  (nearly). 
Can  you  show  why  this  is  true? 

Exercise  40 — Oral. 

Pupils  may  ask  questions  of  their  classmates. 
Teacher  will  select  the  rows.  Be  ready  to  answer  these 
questions  in  good  English. 

1.  How  do  we  find  the  circumference  of  a  circle  when 

we  know  the  diameter? 

2.  How  do  we  find  the  circumference  of  a  circle  when 

we  know  the  radius? 

3.  How  many  degrees  of  arc  are  there  in  every  cir- 

cumference? 

(VIII-71) 


ARITHMETIC 

4.  What  fraction  of  an  entire  circumference  is  an 

arc  of  180°?     Of  90°?     Of  45°? 

5.  Knowing  the  circumference  of  a  circle,  how  can 

we  find  the  length  of  any  arc  of  that  circum- 
ference? 

6.  How  do  we  find  the  area  of  a  circle?     This  sign 

O  is  often  used  to  indicate  a  circle. 

7.  In  what  other  way  can  we  find  the  area  of  a  O  ? 

8.  Knowing  the  area  of  a  O,  how  can  we  find  the 

area  of  any  sector  of  that  circle? 

C  X  R 

9.  Explain  why  ^^^-^ —  =  Area. 

10.  Study  this  out  to  see  if  you  can  tell  why  books 
say  7rR2  =  area  of  a  circle.  Watch  the  substitu- 
tion: 

Area  of  o  =  '^^ 

Area  of  O  = 


Area  of  O  = 


2 
Area  of  O  =  tt  R2. 

Exercise  41 — Written. 

1.  A  circle  has  a  diameter  of  1  yd.;    what  is  the 

length  of  180°  of  its  circumference? 

2.  The  radius  of  a  O  is  2  ft.;   what  is  the  length  of 

60°  of  its  circumference? 

3.  An  arc  of  90°  of  the  circumference  of  a  certain  O 

measures   5.1051    in.    in   length;     what   is   the 
diameter  of  this  O? 

(VIII-72) 


MENSURATION 

4.  An  arc  of  45°  of  the  circumference  of  a  certain  O 

measures  3.1416  in.  in  length;  what  is  the  radius 
of  this  O? 

5.  The  diameter  of  a  certain  circle  is  20  ft. ;  what  is 

the  length  of  360°  of  its  circumference? 

6.  A  certain  O  has  a  radius  of  4  ft.;    what  is  the 

area  of  a  sector  of  this  O  measuring  90°? 

7.  A  certain  circle  has  a  diameter  of  10  ft.;   what  is 

the  area  of  a  sector  of  this  circle  measuring  270°? 

8.  A  O  has  a  circumference  of  314.16  yd.;   what  is 

the  area  of  a  sector  of  this  O  measuring  45°? 

9.  What  is  the  area  of  I  pie  if  the  radius  of  the  pie 

is  5'7 

10.  The  area  of  a  semi-circular  flower  bed  is  25.1328 

sq.  ft. ;  what  is  the  length  of  its  diameter? 

11.  The  area  of  a  certain  quadrant  is  3.1416  sq.  ft.; 

what  is  the  length  of  its  radius? 

12.  The  area  of  a  sector  measuring  72°  of  a   O   is 

392.7  sq.  ft. ;  what  is  the  radius  of  this  O  ? 

13.  The  area  of  a  certain  sector  of  a  circle  is  706.86 

sq.  in.;    if  the  diameter  of  the  circle  is  90  in., 
how  many  degrees  are  there  in  this  sector? 

14.  The  area  of  a  certain  sector  of  a  O  is  508.9392 

sq.  yd.;  the  circumference  of  the  O  is  113.0976 
yd. ;  how  many  degrees  are  there  in  this  sector? 

15.  The  area  of  a  certain  circle  is  31,416  sq.  yd.;  what 

is  the  circumference  of  this  circle? 

16.  A  circular  lake  has  an  area  of  1,256.64  sq.  yd.; 

what  is  the  radius  of  this  lake? 

17.  What  is  the  length  of  90°  of  the  circumference 

of  the  lake  mentioned  in  Question  16? 

(\"III-73) 


ARITHMETIC 


Lesson  21 
Pyramids 


1 


Rectangular 
Prism 


Rectangular 
Pyramid 


Triangular 
Prism 


Triangular 
Pyramid 


Read  carefully. 

A  ''pyramid"  is  a  solid  whose  base  is  a  triangle, 
square,  or  other  polygon,  and  whose  sides  or  lateral 
faces  (corresponding  in  number  to  the  number  of  sides 
in  the  base)  are  triangles  meeting  at  a  common  point 
called  the  vertex. 

If  the  base  of  a  pyramid  is  a  regular  polygon  (that  is, 
a  polygon  whose  sides  are  exactly  equal,  as  an  equi- 
lateral triangle,  a  square,  pentagon,  hexagon,  octagon, 
etc.)  and  the  lateral  faces  are  all  isosceles  triangles, 
then  the  pyramid  is  a  ''regular  pyramid,"  There  are 
other  pyramids,  but  our  lesson  is  confined  to  regular 
pyramids. 

The  distance  from  the  vertex  to  the  center  of  any 
side  of  the  base  is  called  the  "slant  height."  The 
"slant  height"  of  a  pyramid  is  really  the  altitude  of 
one  of  its  faces. 

The  distance  from  the  vertex  to  the  center  of  the 
base  of  the  pyramid  is  the  "altitude."  In  solids,  we 
must  look  very  carefully  for  the  altitude. 

(VIII-74) 


>  =  lateral  area; 


MENSURATION 

The  'lateral  area'^  of  a  pyramid  is  the  sum  of  the 
areas  of  its  lateral  faces.  The  ''entire  area ^'  is  the 
sum  of  the  lateral  area  plus  the  area  of  the  base. 

Since  each  of  the  lateral  faces  of  a  regular  pyramid 
is  an  isosceles  triangle  whose  altitude  is  equal  to  the 
slant  height  of  the  pyramid,  and  whose  base  is  equal 
to  one  side  of  the  base  of  the  pyramid,  therefore: 

I   of  slant  height  XI   side  of  base  =  area  of  1 

lateral  face; 
Area  of  1  lateral  face  X  number ' 
of  lateral  faces 
or 
Perimeter   of   base  X  |   of   slant 

height 

Lateral  area  +  area  of  base  =  entire  area. 
Since  the  volume  of  a  prism  is  found  by  multiplying 
the  area  of  the  base  by  the  altitude,  and  since  it  is  a 
determined  fact  that  the  volume  of  a  pyramid  is  exactly 
i  as  great  as  the  volume  of  a  prism  having  the  same 
base  and  altitude,  therefore: 

Base  X  altitude  -^  3  =  volume    of    pyramid,    or 

B  X  Alt.       ^,  , 
5 =  Volume. 

Exercise  42 — Oral  and  Written. 

Use  the  blackboard  and  scales.  Make  drawings 
rapidly  or  have  them  ready.      Volunteer  to  draw  them. 

1.  What  is  a  polygon  having  three  equal  sides  called? 

Four  equal  sides?     Five?     Six?     Eight? 

2.  Draw  the  five  polygons  referred  to  in  Question  1, 

and  under  each  drawing  write  its  correct  name. 
(VIII-75) 


ARITHMETIC 

3.  A  prism  has  how  many  bases?     A  pyramid  has 

how  many  bases? 

4.  A  triangular  prism  has  how  many  sides  or  lateral 

faces?    What  shape  is  each  of  these  faces? 

5.  A  triangular  pyramid  has  how  many  sides  or  lateral 

faces?    What  shape  is  each  of  these  fac^s? 

6.  A  rectangular  prism  has  how  many  sides  or  lateral 

faces?    What  shape  is  each  of  these  faces? 

7.  A  rectangular  pyramid  has  how  many  sides  or 

lateral  faces?    What  shape  is  each  of  these  faces? 

8.  If  the  base  of  a  prism  is  an  octagon,  how  many 

lateral  faces  has  the  prism?  What  is  the  shape 
of  each  of  the  faces?  If  the  base  is  a  hexagon, 
how  many  lateral  faces  has  the  prism?  What 
shape  is  each  of  the  faces?  If  the  base  is  a 
pentagon,  how  many  lateral  faces  has  the  prism? 
What  shape  is  each  of  the  lateral  faces? 

9.  If  the  base  of  a  pyramid  is  an  octagon,  how  many 

lateral  faces  has  the  pyramid  and  what  is  the 
shape  of  each?  If  the  base  is  a  hexagon?  If 
the  base  is  a  pentagon? 

10.  How  do  we  find  the  area  of  a  A?    How  do  we  find 

the  area  of  one  of  the  lateral  faces  of  a  triangular 
pyramid?     Of  any  other  kind  of  a  pyramid? 

11.  How  do  we  find  the  lateral  area  of  a  pyramid  when 

we  know  the  area  of  1  lateral  face?  How, 
when  we  know  the  perimeter  of  the  base  and 
the  slant  height  of  the  pyramid? 

12.  How  do  we  find  the  entire  area  of  a  pyramid? 

13.  Cut  a  prism  out  of  a  potato  or  make  one  out  of 

clay;  weigh  it  and  note  the  weight. 

CVIII-76) 


MENSURATION 

14.  Without  changing  the  size  of  the  base  or  the 

altitude  of  the  prism  you  have  just  made,  cut  a 
pyramid  out  of  the  prism;  weigh  it  and  note  the 
weight;  also  weigh  the  waste  material  which 
was  cut  away. 

15.  How  does  the  weight  of  the  pyramid  compare 

with  the  weight  of  the  prism? 

Exercise  43 — Class  work  (Construction). 

Everybody  make  these.   Base  2"  X  2\  Altitude  4". 

1.  Make  a  prism  and  a  pyramid  of  heavy  paper, 

using  the  same  base  and  altitude  for  one  that 
you  use  for  the  other;  leave  the  end  of  the 
pyramid  and  one  end  of  the  prism  open.  Fill 
your  prism  full  of  sawdust  by  using  the  pyramid 
as  a  measure;  how  many  times  did  you  have 
to  fill  your  pyramid?  What  is  the  volume  of  the 
prism  as  compared  with  the  volume  of  the 
pyramid? 

2.  Say  how  to  find  the  volume  of  a  pyramid. 

3.  Take  a  prism  in  your  hand  and  say  four  facts 

that  you  have  learned  about  it.  Can  you  give 
more  facts? 

4.  Take  a  pyramid  in  your  hand  and  see  if  you  know 

all  the  interesting  things  about  it  that  you 
should  know.  Point  out  the  important  things 
as  you  proceed. 

5.  How  do  we  find  the  volume  of  any  prism? 

6.  If  the  base  of  a  prism  is  a  2"  square  (we  could  write 

this  2"  n),  and  its  altitude  is  6",  what  is  its 
volume?      If    each    cubic    inch    of    the    prism 
weighed  1  lb.,  what  would  be  its  weight? 
(VIII-77) 


ARITHMETIC 

7.  If  this  prism  was  changed  into  a  pyramid  having 

the  same  altitude  and  base,  what  part  of  its 
volume  would  remain?  What  would  the  pyra- 
mid weigh? 

8.  What  is  the  volume  cf  a  pyramid  whose  base  is  a 

3"  square,  and  whose  altitude  is  10''?  How  do 
we  find  the  volume  of  any  pyramid? 

9.  Draw,  and  state  the  shape  of  the  figure  made  by 

the  three  lines  representing  the  following  three 
dimensions  of  a  pyramid : 

(a)  A  Hne  representing  the  altitude; 

(b)  A  line  representing  the  slant  height; 

(c)  A  line  joining  (a)  and  (b); 

In  the  figure  you  have  drawn,  what  name  is 
given  to  the  line  which  represents  the  slant 
height? 

10.  What  is  the  volume  of  a  prism  whose  base  is  a 

2"  square,  and  whose  altitude  is  12"? 

11.  What  is  the  volume  of  a  pyramid  whose  base  is 

a  2''  square,  and  whose  altitude  is  12"? 

Exercise  44 — Written. 

1.  A  pyramid  has  a  15'  D  for  its  base,  and  its  altitude 
is  20  feet;  what  is  its  volume?  What  is  its 
perimeter?     Its  slant  height? 

2.  The  base  of  a  pyramid  has  the  shape 

of  an  octagon  having  8-in.  sides;  its 

slant  height  is  12  in.;    what  is  its 

lateral  area? 

3.  The  base  of  a  pyramid  is  a  A  having  10"  sides; 

itts  altitude  is  15";   what  is  its  volume?     What 

is  its  slant  height? 

(VIII-78) 


MENSURATION 


4.  How  many  sq.  ft.  of  slate  are  required  to  slate  a 
steeple  having  the  shape  of  a  pyramid  whose 
base  is  a  15'  square,  and  whose  altitude  is  30'? 


5.  The  base  of  a  hexagonal  church  steeple  has  sides 

12'  long;  its  slant  height  is  40';  how  many 
pieces  of  slate  8"  square  will  be  needed  to  slate 
this  steeple,  if  a  6"  square  of  each  piece  of  slate 
is  exposed  to  the  weather? 

6.  What  is  the  volume  of  the  largest  pyramid  which 

can  be  cut  out  of  a  prism  whose  base  is  a  6" 
square,  and  whose  altitude  is  10"? 

7.  The  volume  of  a  pyramid  having  a  square  base  is 

6,250  cu.  ft.;  its  altitude  is  30  ft.;  what  is  the 
length  of  each  side  of  the  base? 

8.  The  volume  of  a  pyramid  having  a  square  base 

is  2,048  cu.  in.;  its  altitude  is  24  in.;  what  is 
its  slant  height? 

9.  The  slant  height  of  a  pyramid  is  5";   its  altitude 

is  4";  its  base  is  a  D  ;  what  is  its  volume? 

(VIII-79) 


ARITHMETIC 


10.  The  Great  Pyramid  of  Cheops  in  Egypt  is  764  ft. 
square  at  the  base,  and  its  altitude  is  480  ft.; 
what  is  its  volume?  What  is  its  slant  height? 
What  is  its  lateral  area? 

Lesson  22 
Cones 


Cylinder  Cone 

Read  carefully. 

A  *^cone"  is  a  solid  having  a  circular  base,  and  one 
curved  side  which  tapers  uniformly  from  the  base  to  a 
vertex  directly  over  the  center  of  the  base. 

The  distance  on  a  straight  line  from  the  vertex  to 
any  point  on  the  circumference  of  the  base  is  the  slant 
height. 

(VIII-80) 


MENSURATION 

The  distance  from  the  vertex  to  the  center  of  the  base 
of  the  cone,  is  the  altitude. 

There  are  cones  of  other  descriptions,  but  they  are 
not  included  in  our  lesson. 

The  lateral  area  of  a  cone  is  the  area  of  its  curved 
surface  or  side.  The  entire  area  of  a* cone  is  the  sum  of 
its  lateral  area  plus  the  area  of  its  base. 

Since  the  lateral  surface  of  a  cone  may  be  regarded 
as  being  an  isosceles  triangle  whose  altitude  is  equal 
to  the  slant  height  of  the  cone  and  whose  base  is  equal 
to  the  circumference  of  the  circular  base  of  the  cone, 
therefore : 

I  of  slant  height  X  circumference  of  base  =  lateral 

area; 
Lateral  area  +  area  of  base  =  entire  area. 

Since  the  volume  of  a  cylinder  is  found  by  multiply- 
ing the  area  of  the  base  by  the  altitude,  and  since,  it 
is  a  determined  fact  that  the  volume  of  a  cone  is  exactly 
^  as  great  as  the  volume  of  a  cylinder  having  the  same 
base  and  altitude,  therefore: 

Area  of  base  X  altitude  -r-  3  =  volume  of  cone. 

Exercise  45 — Oral  and  Written. 

Use  scales,  if  possible. 

1.  Make  a  cone  with  the  same  base  and  altitude  as 

a  cylinder;  fill  the  cylinder  with  sawdust  by 
using  the  cone  as  a  measure.  Is  the  relation  the 
same  as  between  a  prism  and  a  pyramid  of  the 
same  dimensions? 

2.  A  cylinder  has  how  many  bases?    A  cone  has  how 

many  bases? 

(VIII-81) 


ARITHMETIC 

3.  A  cylinder  has  how  many  sides?     Of  what  shape? 

4.  A  cone  has  how  many  sides?     Of  what  shape? 

5.  What  is  the  shape  of  the  curved  surface  of  a 

cyhnder  when  it  is  straightened  out? 

6.  What  does  the  curved  surface  of  a  cone  resemble 

when  it  is  straightened  out? 

7.  How  do  we  find  the  area  of  a  triangle? 

8.  How  do  we  find  the  area  of  the  curved  surface  of 

a  cone? 

9.  How  do  we  find  the  area  of  a  circle  most  quickly 

when  the  circumference  and  diameter  are  known? 

10.  How  do  we  find  the  area  of  a  O  most  quickly 

when  the  radius  is  known?  How  when  the 
diameter  is  known? 

11.  How  do  we  find  the  entire  area  of  a  cone? 

12.  Cut  a  cylinder  out  of  a  potato  or  make  one  out  of 

clay;  weigh  it  and  note  the  weight. 

13.  Without  changing  the  size  of  the  base  or  the  alti- 

tude of  the  cylinder  you  have  just  made,  cut  a 
cone  out  of  the  cylinder;  weigh  it  and  note  the 
weight;  also  weigh  the  waste  material  which 
was  cut  away. 

14.  How  does  the  weight  of  the  cone  compare  with  the 

weight  of  the  cylinder?  How  does  the  weight 
of  the  waste  compare  with  the  weight  of  the 
cylinder? 

15.  How  do  we  find  the  volume  of  any  cylinder? 

16.  If  the  area  of  the  base  of  a  cylinder  is  4  sq.  in., 

and  its  altitude  is  6  in.,  what  is  its  volume?  If 
each  cubic  inch  of  this  cylinder  weighed  1  lb., 
what  would  be  its  weight? 

(VIII-82) 


MENSURATION 

17.  If  this  cylinder  was  changed  into  a  cone  having 

the  same  altitude  and  base,  what  part  of  its 
volume  would  remain?     What  would  it  weigh? 

18.  What  is  the  volume  of  a  cone  whose  base  has  an 

area  of  9  sq.  in.,  and  whose  altitude  is  10  in.? 

19.  How  do  we  find  the  volume  of  any  cone? 

20.  Draw,  and  state  the  shape  of  the  figure  made  by 

the  three  hnes  representing  the  following  three 
dimensions  of  a  cone : 

(a)  A  line  representing  the  altitude; 

(b)  A  line  representing  the  slant  height; 

(c)  A  fine  joining  (a)  and  (h); 

21.  In  the  figure  you  have  drawn,  what  name  is  given 

the  line  which  represents  the  slant  height? 

22.  Take  a  cylinder  in  your  hand  and  tell  the  class 

all  you  know  about  it.  Describe  it  fully  and 
clearly  so  that  those  who  do  not  know  as  well 
as  you  may  learn. 

23.  Take  a  cone  in  your  hand  and  tell  all  about  it. 

Use  good  English  so  that  all  may  understand. 

Exercise  46 — Written. 

1.  A  cone  has  a  20-ft.  circle  for  its  base  and  its  alti- 

tude is  15  ft.;  what  is  the  area  of  its  base? 
What  is  its  volume?  ^^Tiat  is  the  circumference 
of  its  base?    What  is  its  slant  height? 

2.  The  base  of  a  cone  is  a  circle  of  8'^  radius;    its 

slant  height  is  12";  what  is  the  area  of  its  lateral 
surface? 

3.  A  cone  has  a  O  12'Mn  diameter  for  its  base  and 

its  slant  height  is  36";  what  is  its  altitude?/ 

(VIII-83) 


ARITHMETIC 

4.  The  area  of  the  base  of  a  cone  is  28.2744  sq.  in.; 

its  altitude  is  21  in.;  what  is  its  volume?    What 
is  its  slant  height? 

5.  What  is  the  volume  of  the  largest  cone  that  can 

be  cut  out  of  a  cylinder  whose  base  is  4''  in 
diameter,  and  whose  altitude  is  10'7 

6.  What  is  the  volume  of  the  largest  cone  that  can 

be  cut  out  of  a  cylinder  containing  600  cu.  in.? 

7.  The  volume  of  a  certain  cone  is  785.4  cu.  ft.;   if 

its  altitude  is  30  ft.,  what  is  the  diameter  of 
its  base? 

8.  The  volume  of  a  cone  is  37.6992  cu.  ft. ;  its  altitude 

is  4  ft. ;  what  is  its  slant  height? 

9.  The  slant  height  of  a  cone  is  10'';   its  altitude  is 

8";  what  is  its  volume? 
10.  The  circumference  of  the  base  of  a  cone  is  28.2744 
in.;   its  altitude  is  6";   what  is  its  slant  height? 

Lesson  23 

Frustums 
I.  n.  nr  w 


ALTITUDE    /      \  ALTITUDE 


Pyramid  Frustum  of  a  Cone  Frustum  of  a 

Pyramid  Cone 

Read  carefully. 

A  '^frustum"  of  a  pyramid  or  of  a  cone  is  that  part 
of  a  pjnramid  or  of  a  cone  which  lies  between  the  base 
and  a  plane  parallel  to  the  base.     If  you  cut  off  the 

(VIII-84) 


MENSURATION 

upper  part  of  a  pyramid  or  of  a  cone  so  that  the  cut  is 
parallel  to  the  base,  the  remaining  part  is  a  frustum. 

The  surface  between  the  two  bases  is  the  lateral 
surface. 

The  area  of  the  lateral  surface  is  the  lateral  area. 

The  lateral  area  plus  the  areas  of  the  two  bases,  is 
the  entire  area. 

The  distance  between  the  centers  of  the  two  bases  is 
the  altitude. 

How  would  you  find  the  area  of  the  frustum  in 


] 


Figure  II?  i — 

Open  it  out  on  the  board :      1 

How  would  you  find  the  area  of  the  frustum  in 
Figure  IV?                                 r— 
Open  it  out  on  the  board :    L 

To  find  the  lateral  area  of  a  frustum,  find  the  average 
of  the  perimeters  of  the  two  bases  by  adding  the  two 
perimeters  and  dividing  the  sum  by  2,  then  multiply 
this  average  perimeter  by  the  slant  height.  Can  you 
see  how  this  simplifies  your  process? 

Exercise  47 — Oral. 

Some  boy  may  draw  these  on  the  board  for  the  class 
before  class  time. 

1.  Can  you  make  this  into  a  rect-  'Q 

angle?  What  are  the  dimensions      /        i^      \ 
of  the  rectangle?  '2.' 

2.  Can  you  make  a  rectangle  of  it  by  using  the 
I — ^  average    line    between    the    two 

x-nM       Ir^x     bases?      Where   must    the    two 

pieces  marked  ''X"  be  placed? 
(VIII-85) 


ARITHMETIC 

3.  Mention  some  things  you  know  which  are  formed 

Hke  the  frustum  of  a  cone.  Mention  some  which 
are  formed  hke  the  frustum  of  a  pyramid. 
(Think  of  any  long  walk  you  have  taken.) 

4.  How  many  bases  has  the  frustum  of  a  pyramid? 

Of  a  cone? 

5.  The  number  of  lateral  faces  of  the  frustum  of  a 

pyramid  depends  on  what? 

6.  How  many  lateral  faces  has  the  frustum  of  a  cone? 

7.  What  constitutes  the  lateral  surface  of  the  frustum 

of  a  pyramid?     Of  a  cone? 

8.  What  constitutes  the  lateral  area  of  the  frustum 

of  a  pyramid?     Of  a  cone? 

9.  What  constitutes  the  entire  area  of  the  frustum 

of  a  pyraimd?     Of  a  cone? 

10.  From  what  point  to  what  point  is  the  slant  height 

of  the  frustum  of  a  pyramid  measured?  Of  a 
cone? 

11.  From  what  point  to  what  point  is  the  altitude 

of  the  frustum  of  a  pyramid  measured?  Of  a 
cone? 

12.  If  one  of  the  lateral  faces  of  the  frustum  of  a 

pyramid  measures  4''  in  width  along  the  upper 
base,  and  6''  in  width  along  the  lower  base,  at 
what  point  between  the  two  bases  would  it 
measure  5"  in  width?  What  would  be  the 
average  width  of  this  face?  If  this  face  were  5'' 
in  width  at  all  points  between  the  two  bases 
what  would  be  its  shape?  How  would  its  area 
be  found?  How  do  you  find  the  area  of  one  of 
the  lateral  faces  of  the  frustum  of  a  pyramid? 
(Vlll-Sfi) 


MENSURATION 

13.  How  do  you  find  the  average  perimeter  of  the 
bases  of  any  frustum?  How  do  you  find  the 
area  of  the  lateral  surface  of  any  frustum? 

Exercise  48 — Written. 

Make  large  excellent  drawings  before  you  begin  your 
calculations. 

1.  A  round  tin  pail  measures  16''  in  diameter  at  the 

top,  and  10"  in  diameter  at  the  bottom;  its  slant 
height  is  12'';  how  much  tin  is  there  in  the  side 
of  the  pail?  How  much  in  the  bottom?  How 
much  in  the  entire  pail? 

2.  The  dimensions  of  a  photographer's  tray  are  10" 

X  10"  at  the  top,  and  8"  X  8"  at  the  bottom, 
and  the  slant  height  is  2";  what  is  the  area  of 
the  sides  and  bottom  of  this  tray? 

3.  The  diameter  of  a  flower  pot  is  10"   at  the  top, 

and  6"  at  the  bottom;  its  slant  height  is  12"; 
what  is  the  area  of  its  lateral  surface? 

4.  A  circular  pan  measures  20"  in  diameter  at  the 

top  and  8"  in  diameter  at  the  bottom;  its  slant 
height  is  10";  what  is  the  area  of  the  material 
out  of  which  it  is  made? 

5.  The  area  of  the  material  in  a  pan  is  208  sq.  in.; 

it  measures  10"  X  10"  at  the  top  and  8"  X  8" 
at  the  bottom;   what  is  its  slant  height? 

6.  The  frustum  cf  a  pyramid  has  for  its  lower  base 

an  equilateral  A  whose  sides  measure  6",  and 
for  its  upper  base  an  equilateral  A  whose  sides 
measure  4";    the  slant  height  of  the  frustum  is 
10";  what  is  its  lateral  area? 
(VIII-87) 


ARITHMETIC 


7.  The   lateral   surface   of  the   frustum   of  a   cone 

measures  565.488  sq.  in.;  the  diameter  of  its 
upper  base  is  10'';  its  slant  height  is  12'^;  what 
is  the  diameter  of  its  lower  base? 

8.  The   lateral   surface   of   the   frustum   of  a   cone 

measures  311.0184  sq.  ft. ;  the  radius  of  its  upper 
base  is  4  ft. ;  the  radius  of  its  lower  base  is  5  ft. ; 
what  is  its  slant  height? 

9.  The  upper  base  of  the  frustum  of  a  pyramid  is  a 

10''  D ;   its  lower  base  is  a  22"  D ;   its  altitude 
is  8";  what  is  its  slant  height? 
10.  The  upper  base  of  the  frustum  of  a  cone  is  6"  in 
diameter;    the  lower  base  is  60"  in  diameter; 
its  slant  height  is  45";  what  is  its  altitude? 

Lesson  24 
Spheres 


Read  carefully. 

A  ''sphere"  is  a  round  solid  bounded  by  a  uniformly 
curved  surface,  every  point  of  which  is  equally  distant 
from  a  point  within,  called  the  center. 

The  circle  made  by  cutting  a  sphere  into  two  equal 
parts  or  hemispheres,  by  a  plane  passing  through  the 
center,  is  called  a  ''great  circle"  of  the  sphere.  Name 
a  great  circle  of  the  earth. 

(VIII-88) 


MENSURATION 

The  distance  from  surface  to  surface  through  the 
center  is  the  diameter  of  the  sphere,  and  is  the  same  as 
the  diameter  of  the  great  circle. 

The  distance  from  the  center  to  any  point  on  the 
surface  is  the  radius  of  the  sphere,  and  is  the  same  as  the 
radius  of  the  great  circle. 

The  circumference  of  a  sphere  is  the  same  as  the  cir- 
cumference of  its  great  circle.  The  circumference  is 
the  longest  curved  line  around  the  sphere. 

By  placing  a  tack  in  the  center  of  the  curved  surface 
of  one  of  the  hemispheres,  and  another  tack  in  the 
center  of  the  flat  surface  of  the  other  hemisphere,  and 
winding  cord  carefully  around  each  to  cover  the  two 
surfaces,  you  will  find  that  exactly  twice  as  much  cord 
is  requu'ed  to  cover  the  curved  surface  as  is  required 
to  cover  the  flat  surface;  hence,  the  area  of  the  curved 
surface  of  the  entire  sphere  equals  four  times  the  area 
of  its  great  circle.  The  area  of  the  great  circle  equals 
ttR-;  therefore,  the  area  of  the  curved  surface  equals 
47rR^.  You  can  also  prove  this  by  cutting  several 
circles  out  of  paper;  make  the  circles  the  same  diameter 
as  the  diameter  of  a  wooden  ball  (you  can  cut  an  old 
croquet  ball  in  half);  tear  one  circle  at  a  time  into 
small  pieces  and  paste  to  cover  the  ball.  How  many 
circles  are  needed? 

If  a  sphere  were  cut  into  small  pjrramids  as  is  sho\\Ti 
in  the  illustration,  each  of  the  pyramids  would  have  an 
altitude  equal  to  the  radius  of  the  sphere,  and  the  com- 
bined area  of  the  bases  of  the  small  pyramids  would 
be  approximately  the  same  as  the  area  of  the  surface 
of  the  sphere.     Since  we  find  the  volume  of  a  pjo-amid 

(VIII-89) 


ARITHMETIC 

by  multiplying  the  area  of  its  base  by  J  the  altitude, 
we  can  find  the  volume  of  a  sphere  by  multiplying  the 

area  of  its  curved  surface  by  J  its  radius.     Therefore, 

R  4 

since  47rR2  =  area, -^  X  47rR^or^  ttR^,  =  volume. 

Exercise  49 — Oral. 

Choose  one  of  your  classmates  to  answer.  His  row 
asks  him  any  of  these  questions.  After  losing  or 
answering  successfully  five  of  these,  another  row  takes 
it  up,  etc. 

1.  Name  several  things  you  know  which  are  spheres 

in  shape. 

2.  When  a  sphere  is  cut  into  two  equal  parts  by  a 

plane  passing  through  the  center,  what  is  each 
of  the  two  parts  called? 

3.  What  name  is  given  to  the  circular  plane  surface 

made  by  cutting  a  sphere  into  tw^o  equal  parts? 

4.  How  is  the  diameter  of  a  sphere  measured?     How 

the  radius?     How  the  circumference? 

5.  With  what  measurement  of  the  great  circle  of  a 

sphere  does  the  diameter  of  the  sphere  corres- 
pond?    The  radius?     The  circumference? 

6.  Does  it  require ,  more  cord  to  cover  the  curved 

surface  or  the  flat  surface  of  a  hemisphere? 
How  many  times  as  much?  How  many  times 
as  much  is  required  to  cover  the  curved  surfaces 
of  both  hemispheres  as  is  required  to  cover  the 
flat  surface  of  one  hemisphere? 

7.  What  is  the  ratio  of  the  area  of  the  curved  surface 

of  a  sphere  to  the  area  of  the  flat  surface  of  one 
of  its  hemispheres? 

(\aiI-90) 


I 


MENSURATION 

8.  How  do  we  find  the  area  of  any  circle?  How  do 
we  find  the  area  of  the  great  circle  of  a  sphere? 
How  do  we  find  the  area  of  the  cuin^ed  surface 
of  a  sphere? 

9.  If  a  sphere  is  cut  into  small  pyramids  as  is  shown 
in  the  illustration,  what  dimension  of  the  sphere 
would  correspond  to  the  altitude  of  each  of  the 
pyramids? 

10.  Which  dimension  of  the  sphere  would  correspond 

to  the  combined  area  of  the  bases  of  all  of  the 
small  pyramids? 

11.  How  do  we  find  the  volume  of  any  pyramid? 

12.  If  we  know  the  radius  of  a  sphere  and  the  area  of 

its  curved  surface,  how  can  we  find  the  volume? 

13.  Since  ttR^  =  area  of  any  circle,  state  in  the  form 

of  an  equation  the  formula  for  finding  the  area 
of  the  curved  surface  of  a  sphere. 

14.  What  change  do  you  make  in  your  equation  to 

show  volume? 

15.  Given  the  radius  of  a  sphere,  state  the  shortest 

way  of  finding  its  volume. 

16.  Take  a  ball  and  tell  all  the  rules  you  know  about  it. 

Exercise  50 — Written. 

1.  The  radius  of  a  sphere  is  10'';  what  is  the  area  of 

its  great  circle?     W^hat  is  the  area  of  its  curved 
surface? 

2.  The  diameter  of  a  sphere  is  14  ft.;    what  is  the 

area  of  its  curved  surface? 

3.  What  is  the  area  of  the  cover  of  a  baseball  2  J''  in 

diameter? 

(VIII-91) 


ARITHMETIC 

4.  The  diameter  of  the  earth  is  approximately  8,000 

miles;  what  is  the  area  of  the  earth's  surface? 

5.  The  outside  of  the  dome  of  an  astronomical  observ- 

atory is  in  the  form  of  a  hemisphere  50  ft.  in 
diameter;  at  25^^  per  square  yard,  what  would 
be  the  cost  of  painting  the  outside  of  this  dome? 

6.  The  area  of  the  surface  of  the  moon  is  about 

12,566,400  sq.  mi. ;  what  is  the  area  of  its  great 
circle?  What  is  the  length  of  its  radius?  What 
is  the  length  of  its  diameter? 

7.  What  is  the  volume  of  an  orange  whose  area  is 

50.25  sq.  in.  and  whose  radius  is  2"? 

8.  What  is  the  volume  of  the  largest  sphere  that  can 

be  turned  from  a  6"  cube  of  wood? 

9.  Circular  discs  of  flat  sheet  metal  are  often  pressed 

by  machinery  into  the  form  of  hemispheres. 
The  area  of  the  circular  disc  must  equal  the 
area  of  the  finished  hemisphere.  What  is  the 
diameter  of  the  disc  required  to  make  a  hemi- 
sphere whose  diameter  is  12"? 
10.  What  would  be  the  diameter  of  the  sphere  which 
could  be  made  by  joining  the  two  hemispheres 
pressed  out  of  two  brass  discs  each  10"  in 
diameter?  What  would  be  the  volume  of  this 
sphere. 

Exercise  51 — Oral  Review. 

1.  How  many  degrees  of  arc  are  there  in  a  circle? 
In  a  semicircle?     In  a  quadrant?     From  the 
equator  to  the  North  Pole?     From  the  North 
Pole  to  the  South  Pole? 
(VIII-92) 


MENSURATION 

2.  How  many  degrees  are  there  in  a  right  angle? 

3.  How  much  time  corresponds  to  180"^  of  longitude? 

How  much  time  corresponds  to  90°?    15°?    1°? 

4.  Whsit  is  the  unit  of  length  in  the  metric  system? 

Of  volume?      AYeight?      Capacity?      Area? 

5.  What    single    word   means    100  m.?      10,000  L? 

1,000  g.?    10  g.?     Im.?    .11?    .01m.?    .001  g.? 

6.  How  do  we  find  the  area  of  a  triangle?    Of  a  circle? 

Of  a  trapezoid? 

7.  How  do  we  find  the  area  of  the  lateral  surface  of 

a  cylinder?     Of  a  cone?     Of  a  prism?     Of  a 
pyramid? 

8.  How  do  we  find  the  volume  of  a  cube?     Of  a 

prism?     Cylinder?     Pyramid?     Cone? 

9.  How  do  we  find  the  circumference  of  a  circle? 

10.  How  do  we  find  the  perimeter  of  the  base  of  a 

pyramid?     Of  a  prism?     Cylinder?     Cone? 

11.  How  do  we  find  the  average  perimeter  of  the  bases 

of  the  frustum  of  a  pyramid? 

12.  How  do  we  find  the  average  perimeter  of  the  bases 

of  the  frustum  of  a  cone? 

^Add  the  following: 


13. 

14. 

15. 

16. 

46,314 

39,876 

35,212 

56,598 

38,726 

67,968 

64,911 

67,873 

34,912 

46,436 

17,878 

43,289 

62,848 

34,999 

43,219 

21,196 

36,944 

89,722 

38,777 

55,439 

56,497 

32,464 

58,649 

58,935 

(VIII-93) 


ARITHMETIC 
Exercise  52 — AVritten  Review. 


1.  V2,371.69  =  ? 

2.  394'^  =  ? 


-i        -.72  8  9     =     ? 
'^'       V    361 

4.  What  is  the  hypotenuse  of  a  right  triangle  whose 

legs  measure  respectively  15  ft.  and  20  ft.? 

5.  An  equilateral  triangle  has  sides  measuring  8  in.; 

what  is  its  altitude? 

6.  An  arc  of  45°  of  the  cuxumference  of  a  certain 

circle  measures  4.7124''  in  length;    what  is  the 
diameter  of  this  circle? 

7.  The  area  of  a  certain  sector  of  a  circle  is  52.36 

sq.  in.;   if  the  diameter  of  this  circle  is  20  in., 
how  many  degrees  are  there  in  this  sector? 

8.  The  volume  of  a  p^Tamid  having  a  square  base  is 

48  cu.  in.;   its  altitude  is  4'^;    what  is  its  slant 
height? 

9.  The  slant  height  of  a  cone  is  36";    the  radius  of 

its  base  is  6'';  what  is  its  altitude? 
10.  The  area  of  a  sphere  is  2,123.7216  sq.  in.;  what  is 
its  radius? 

Copy  and  multiply: 

(Time  for  these  8  examples  is  less  than  4  minutes.) 


15.  7,326  X  38; 

16.  4,783  X  49; 

17.  3,234  X  98; 

18.  6,218  X  56. 


11.  1,874  X  45 

12.  3,692  X  63 

13.  8,741  X  92 

14.  5,039  X  27: 

Copy  and  divide: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 
19.  49,786  -^  73;  20.  15,652  --  28; 

(VIII-04) 


MENSURATION 

21.  18,009  -^  23;  23.  44,541  -^  63; 

22.  22,225  -v-  25;  24.  29,045  -^  37. 

Add,  but  do  not  copy:     , 

(Time  for  these  6  examples  is  less  than  4  minutes.) 


25. 

26. 

27. 

28. 

29. 

30. 

5,419 

8,732 

9,328 

7,514 

3,928 

4,609 

7,429 

1,872 

4,687 

3,975 

2,448 

6,987 

3,921 

8,728 

6,215 

4,988 

7,397 

5,025 

9,627 

4,187 

8,199 

2,875 

5,622 

1,987 

2,639 

4,879 

1,628 

5,319 

7,875 

5,188 

6,395 

2,486 

3,798 

6,295 

6,894 

9,127 

Subtract,  but  do  not  copy: 

(Time  for  these  12  examples  is  less  than  4  minutes.) 

31.                   32.  33.                   34. 

863,472          749,802  621,804  831,841 

519,728          358,473  239,758  293,957 

35.                  36.  37.                  38. 

612,041          541,387  716,284  300,412 

309,409          297,492  591,876  275,583 

39.                   40.  41.                   42. 

975,319    716,304  824,508  612,387 

493,941    593,295  713,499  509,368 


(VIII-95) 


GRAPHIC    CHARTS   AXD    METERS 


T  ^cc.-.AT  9=; 


Graphic  Charts 

GrspMc  charts  or  graphs"  as  they  are  frequently 
eaQed;  can  be  loade  in  a  great  variety  of  ways  and  can 
be  used  ::r  zi  irpoees.       In  short,  any  drawing 

or  chart  which  shows  at  a  gJance  comparative  statistics 
cr  o'  be  called  a  "graph.'' 

To  snow  alive  production  of  a  certain 

connnodity  lor  various  periods  of  time,  or  for  various 
sections  c*f  the  c  r~:-  7':torial  graphs"  are  often 
used  see  Fir  re  1  7_-  -:yle  of  graph  is  mo<st  useful 
^hen  the  r:___   iirv  can  be  faithfullv  oortraved. 


HAY 


11%^ 


New  Yflfk  =  5-5S>MjOO  T.  I .  :  ^  =  o-S  Imj  1 . 

Hay  Prodsietiicii-  1914 

Tisure  1> 

TIIt^6) 


GRAPHIC   CH_1ETS   AXD   ALETERS 


To  record  variations  within  definitely  known  limits, 
tne  '"line  graph"  is  most  generally  used  Figure  2). 
This  style  of  graph  is  convenient  for  r' -^iiir  r  "^  rres 
in  prices ;  in  prcwinction.  in  temperarLj  ^ .  .:  . .  ^i^e 

by    which    the    desired    information    is    :   _ 
freouentlv  called  the  "curve." 


r-d     is 


I    ^    i 

•i-    1^    - 


73' 


S.       t-       >        o 
IV       0       Z       tl 

;5       0       Z       n 


L5' 


T;.e 


^e  in  CMcago  fa*  s  Yesr 


This  line  graph  shows  two  separate  kinds  of  inf<»- 
mation.  The  continiious  line  or  "curve*'  sho^^;^  the 
"mean"  or  ocf:^  :J  average  temperature  for  the  twelve 
months  of  a  certain  year,  while  the  dotted  line  or 
"curve"  shows  the  "normal"  or  usu^I  average  temper- 
ature for  the  corresponding  months  oi  preceding  years. 

For  almost  ever^'  purpose  involving  comparisons, 
the  '"bar  graph"  is  adaptable  Figure  3\  The  bar 
graph  and  the  line  graph  are  more  frequently  used  than 
any  other  forms  oi  graphs. 


ARITHMETIC 


MILLION  TONS 

so      100     ISO    200    250    300    350  400   450    SCO    550   600 


I9IO 

1311 

I9IE 

1913 

1914 

1915 


447,853,999.  T 
443,054.614  T 
477,202.303  T 
508,970.640.  T 
-458.50489aT. 
47-4.660.256.T. 


Coal  Production  of  United  States,  1910-1915. 
(Figure  3) 

To  illustrate  percentages,  the  '^circle  graph"  is  very 
useful  (Figures  4,  5,  and  6).  In  this  style  of  graph, 
the  complete  circle  represents  100%,  and  the  relative 
sizes  of  the  various  sectors  show  the  comparative  per- 
centages. 


World's  Cotton 

Production,  1913 

(Figure  4) 


Area  of  U.  S.  Land 
(Figure  5) 


Value  of  1909  U.  S. 

Mining  Products 

(Figure  6) 


Exercise  53 — Oral. 

1.  What  information  is  shown  by  the  graph  in  Figure 

1?     Read  this  graph. 

2.  What  information  does  the  continuous  curve  in 

Figure  2  show?     What  information  is  shown  by 

(VIII-98) 


11 


GRAPHIC  CHARTS  AND  METERS 

the  dotted  line?   Read  this  graph.    AY  hen  would 
you  choose  to  use  this  one? 

3.  What  is  compared  by  the  graph  in  Figure  3? 

From  this  graph,  compare  the  coal  production 
of  1912  with  that  of  1915. 

4.  Tell  about  the  graph  in  Figure  4. 

5.  Explain  Figure  5. 

6.  AYhat  information  do  you  see  in  the  graph  in 

Figure  6?   Read  some  relations  found  in  this  one. 

7.  Which  style  of  graph  would  you  prefer  to  use,  if 

you  wanted  to  show  the  apple  production  of 

Michigan  in  comparison  with  that  of  Illinois? 

Exercise  54 — Written. 


L       1.  What  is  the  ratio  of  the  hay  production  in  Florida 
to  the  hay  production  in  New  York  as  found  by 

I  a  comparison  of  the  areas  of  the  two  drawings 

in  Figure  1? 
2.  Draw  a  pictorial  graph   showing   that   in   1914 
Pennsylvania  produced  about  148,000,000  tons 
of  coal,  while  Oklahoma  produced  about  4,000,- 
ll  000  tons. 

3.  Draw  a  line  graph  showing  that  the  price  of  wheat 
on  October  1st  of  each  year  from  1910  to  1918 
inclusive  was  as  follows: 

1910 $0.94  1914 $0.94 

1911 0.88  1915 0.91 

1912 0.83  1916 1.36 

1913 0.78  1917 .   2.01 

1918 $2.06 

CV'III-99) 


ARITHMETIC 

4.  Draw  a  line  graph  showing  (by  a  continuous  line) 

the  membership  of  a  certain  club  on  the  first  day 
of  each  month  during  the  year  1918,  and  show 
'  also  on  the  same  graph  (by  a  dotted  line)  the 
membership  on  the  first  day  of  the  correspond- 
ing month  in  previous  years : 

Membership 
Date  Membership,  1918        Previous  Years 

Jan.    1st 410  390 

Feb.   1st 430  420 

Mar.  1st 435  425 

Apr.   1st 440  430 

May  1st 430  440 

Jun.    1st 425  435 

Jul.     1st 420  430 

Aug.  1st 415  420 

Sep.    1st 410  410 

Oct.    1st 415  405 

Nov.  1st 420  400 

Dec.   1st 425  395 

5.  Draw  a  bar  graph  to  show  the  population  of  the 

United  States  by  millions  for  each  census  year 
from  1850  to  1910: 

Census  Year  Population 

1850 23,000,000 

1860 31,000,000 

1870 39,000,000 

1880 50,000,000 

1890 63,000,000 

1900 76,000,000 

1910 92,000,000 

(VIII-lOO) 


GRAPHIC  CHARTS  AND  METERS 

6.  Draw  a  bar  graph  showing  the  steel  production 

of  the  United  States  by  milHons  of  tons  for  each 

year,  1906  to  1916: 

Year  Tons  of  Steel 

1906 23,000,000 

1907 23,000,000 

1908 14,000,000 

1909 24,000,000 

1910 26,000,000 

1911 24,000,000 

1912 31,000,000 

1913 31,000,000 

1914 24,000,000 

1915 32,000,000 

1916 43,000,000 

7.  Draw  a  circle  graph  showing  the  percentage  of  the 

population  distributed  in  each  of  the  following 
divisions : 

(Use  your  protractor  to  divide  the  circle  into 
sectors  containing  the  correct  number  of  degrees.) 
Division  Percentage 

New  England 7. 

Middle  Atlantic 21. 

East  North  Central 20. 

West  North  Central 13. 

South  Atlantic 13. 

East  South  Central 9. 

West  South  Central 10. 

Mountain 3. 

Pacific 4. 

Total ioa 

(VIII-lOl) 


ARITHMETIC 

8.  Draw  a  circle  graph  showing  the  monthly  sales 
of  a  large  wholesale  house  to  be  divided  as 
follows : 

Commodity  Sales 

Dry  Goods $415,000. 

Furniture 1,037,500. 

Carpets 352,750. 

Curtains 103,750. 

Miscellaneous 166,000. 

Total $2,075,000. 


Lesson  26 

Meters 

Read  very  carefully. 

The  unit  used  for  the  measurement  of  electrical 
energy  is  the  ^'kilowatt,"  and  the  special  working  unit 
in  commercial  use  is  that  indicating  the  continuous  use 
of  1  kilowatt  of  energy  for  1  hour  of  time,  and  is  called 
the  ''kilowatt  hour"  (k.  w.  hr.). 

Electricity  is  sold  to 
the  consumer  on  the 
basis  of  a  certain  rate 
per  kilowatt  hour,  and  a 
meter  is  used  to  record 
the  number  of  kilowatt 
hours  to  be  charged. 
The  dial  of  the  electric 
meter  here  illustrated 
shows  more  than  3  hun- 
dred-thousands,       more 

The  Dial  of  an  Electric  Meter  than      6      ten-thoUSands, 

(VIII-102) 


GRAPHIC  CHARTS  AND  METERS 

more  than  8  thousands,  more  than  5  hundreds,  1  ten 
and  4  units  of  kilowatt  hours;  to  be  exact,  this  meter 
records  368,514  kilowatt  hours. 

To  ascertain  the  consumption  of  electricity  for  any 
given  period  of  time,  the  meter  reading  at  the  beginning 
of  the  period  is  subtracted  from  the  meter  reading  at 
the  end  of  the  period,  the  difference  between  the 
two  readings  being  the  consumption  for  the  period. 

The  consumption  of 
gas  is  measured  by 
cubic  feet.  The  gas, 
after  being  manufac- 
tured, is  stored  in  large 
tanks  called  '^gas 
holders'^  from  which  it 
is  forced  under  press- 
ure into  the  mains.  From  the  gas  mains  it  passes 
through  the  meter  (where  it  is  measured)  into  the  gas 
pipes  on  the  consumer's  premises. 

The  dial  of  the  gas  meter  here  illustrated  shows  more 
than  7  ten-thousands,  more  than  8  thousands,  and  5 
hundreds  of  cubic  feet,  the  exact  meter  reading  being 
78,500  cu.  ft.  Gas  is  charged  to  the  consumer  at  a 
certain  rate  per  1,000  cu.  ft.  and  the  quantity  consumed 
is  determined  by  subtraction  in  the  same  manner  as  in 
the  case  of  an  electric  meter. 

Exercise  55 — Oral. 

Some  boy  may  draw  dials  of  both  meters  on  the 
board.  Choose  one  of  your  classmates  to  be  the  '^gas 
man"  or  ^'electric  man"  who  comes  to  read  the  meter. 

(VIII-103) 


The  Dial  of  a  Gas  Meter 


ARITHMETIC 

Some  girl  may  change  the  hands  on  the  meters  for  him 
to  read.     Ask  the  following  questions: 

1.  What  is  meant  by  a  kilowatt  hour? 

2.  Why  do  electric  lighting  companies  install  meters 

on  the  premises  of  their  patrons? 

3.  What  do  such  meters  show? 

4.  How  can  we  ascertain  the  quantity  of  electrical 

energy  to  be  charged  for  a  certain  period? 

5.  An   electric   meter   shows   4,319   kilowatt   hours 

consumption;  between  what  two  figures  is  the 
hand  on  the  thousands'  dial?  On  the  hundreds' 
dial?     On  the  tens'  and  units'  dial? 

6.  What  unit  is  used  for  the  measurement  of  gas? 

7.  How  does  a  gas  company  measure  the  quantity 

of  gas  used  by  each  of  its  patrons? 

8.  The  rate  charged  for  gas  is  based  on  what? 

9.  The  reading  on  a  gas  meter  on  Nov.  6th  was  7,000; 

on  Dec.  6th  it  was  9,000;  what  information  can 
you  obtain  from  this  data? 

10.  Read  the  two  meters  illustrated  in  Lesson  26. 

11.  Study  your  gas  and  electric  meters  at  home;  copy 

the  position  of  the  hands  on  a  piece  of  paper 
and  bring  it  to  school;  be  ready  to  read  the 
meters. 

Exercise  56 — Written. 

Office  and  officials  are  to  be  in  one  part  of  the  room. 
Go  to  the  office  to  pay  bills.  Ask  questions  if  you 
want  information  about  your  bills. 

1.  The  reading  of  an  electric  meter  on  Jan.   10th 
was  1,624;    on  Feb.  10th  it  was  1,648;    at  lOf^ 
(VIII-104) 


GRAPHIC  CHARTS  AND  METERS 

per  k.  w.  hour,  what  was  the  amount  of  the  bill 
rendered  on  Feb.  10th? 

2.  A  discount  of  1|^  per  k.  w.  hr.  is  allowed  on  the 

bill  in  Question  1  for  prompt  payment  on  or  be- 
fore Feb.  20th;  what  was  the  net  amount  paid  if 
this  bill  was  settled  Feb.  19th? 

3.  A  certain  lighting  company  charges  lOjz^  per  k.  w. 

hr.  for  the  first  30  k.  w.  hr.  and  Q^  per  k.  w.  hr. 
for  all  in  excess  of  30  k.  w.  hr.,  and  allows  a  dis- 
count of  Iji^  per  k.  w.  hr.  on  the  10^  portion  of 
the  bill  for  prompt  payment;  what  is  the  gross 
amount  of  the  bill  rendered  July  15th  if  the 
meter  reading  was  3,978  on  June  15th  and  4,036 
on  July  15th?   What  was  the  net  amount  paid? 

4.  If  the  reading  of  the  meter  in  Question  3  was  4,212 

on  July  15th,  what  was  the  gross  amount  of  the 
bill  for  the  month  ending  July  15th?  What  was 
the  net  amount  paid  in  settlement  of  this  bill? 

5.  An  electric  lighting  company  charges  11^^  per  k.  w. 

hr.  for  the  first  40  k.  w.  hr.,  7^  for  the  second 
40  k.  w.  hr.,  and  4^  for  all  in  excess  of  80  k.  w. 
hr.;  this  company  allows  a  discount  of  IJ^  per 
k.  w.  hr.  on  the  llf^  and  7^  portions  of  the  bill; 
one  of  its  meters  read  2,611  on  Sept.  20th,  and 
2,706  on  Oct.  20th;  what  was  the  gross  amount 
of  the  bill  rendered  Oct.  20th?  What  was  the 
net  amount  paid  in  settlement  of  this  bill? 

6.  If  the  reading  of  the  meter  in  Question  5  was  2,638 

on  Sept.  20th,  what  was  the  gross  amount  of  the 
bill  for  the  month  ending  Oct.  20th?     What  was 
the  net  amount  paid  in  settlement  of  the  bill? 
CVlII-105) 


ARITHMETIC 

7.  A  gas  company  charges  37^  for  the  first  350  cu.  ft. 

consumed,  and  88^  per  thousand  for  the  excess 
over  350  cu.  ft.;  one  of  its  meters  read  9,500 
Oct.  8th  and  11,300  Nov.  8th;  what  was  the 
amount  of  the  bill  rendered  Nov.  8th? 

8.  If  the  rate  in  Question  7  were  90^  per  thousand  cu. 

ft.  with  a  discount  of  10^  per  thousand  cu.  ft.  for 
prompt  payment,  what  would  be  the  gross 
amount  of  the  bill?  What  would  be  the  net 
amount  paid  in  settlement  of  the  bill? 

9.  Draw  the  dials  of  an  electric  meter  reading  8,643. 
10.  Draw  the  dials  of  a  gas  meter  reading  82,700. 


(vni-106) 


PERCENTAGE 

Lesson  27 

Interest  on  Installment  Accounts 

Read  all  of  this  very  carefully. 

You  have  already  learned  that  since  we  charge 
interest  on  a  sum  for  only  such  time  as  the  sum  remains 
unpaid,  all  partial  pajments  on  the  principal  must 
naturally  affect  the  amount  of  interest,  as  theprincipal 
will  change  as  many  times  as  there  are  partial  payments. 

It  is  customary  that  when  goods  are  purchased  on 
the  installment  plan,  that  is,  under  an  agreement  to 
make  a  certain  number  of  equal  partial  pa}'ments  at 
certain  equal  interv^als  of  time  Tusually  one  pajTnent 
per  month),  interest  is  charged  on  the  diminishing 
balance  from  month  to  month,  and  is  paid  each  month 
when  the  partial  pajmient  is  made  on  the  principal. 

While  interest  of  this  kind  can  be  computed  in  the 
usual  manner  by  making  as  many  interest  calculations 
as  there  are  partial  payments,  much  time  and  labor  can 
be  saved  by  using  averages  in  problems  of  this  kind. 

Practice  Exercise: 

1  month  =  what  part  of  a  year? 
6%  per  annum  =  ?  %  for  1  month? 
At  6'^  per  annum,  what  is  the  interest  for  1  month 
on:  SIOO.  $150.  S75.  S25. 

S200.  S2-40.  S60.  S20. 

$300.  $:360.  $40.  $10. 

r^T[T-107) 


ARITHMETIC 


EXAMPLE:  A  parlor  suite  was  bought  for  $150.00  on  terms  of 
$30.00  cash  and  $10.00  per  month  with  interest 
at  6%  per  annum;  what  was  the  interest  on 
this  account? 

Total  Sale $150.00 

Cash  Paid 30.00 

$120.00  =  12  Payments  @  $10.00  each. 
6%  Int.  for  1  mo.  on  $120.00  (largest  interest  payment)  =  $0.60 
6%  Int.  for  1  mo.  on      10.00  (smallest  interest  payment)   =    0.05 

$065 
Average  interest  payment  =  f  of  $0.65,  or  $0.32^ 
12  Interest  payments  @  $0.32|  each  =  $3.90,  Total  Interest. 


To  use  averages  in  figuring  interest  on  installment 
accounts,  we  find  the  interest  to  be  paid  with  the  first 
installment  (that  being  the  largest  interest  payment), 
then  we  find  the  interest  to  be  paid  with  the  last  install- 
ment (that  being  the  smallest  interest  payment),  then 
we  add  these  two  amounts  of  interest  and  divide  the 
sum  by  2  to  find  the  average  interest  payment  and 
multiply  this  average  by  the  number  of  installments  to 
be  made;  the  answer  so  found  is  the  interest  on  the 
entire  account. 

Always  figure  the  interest  at  the  rate  of  6%  per  annum 
by  using  the  "1%  60-day''  method  (J%  per  month), 
and  when  the  rate  is  other  than  6%,  reduce  your  total 
interest  to  such  other  rate  by  the  use  of  fractions,  as : 
I  for  5%,  i  for  7%,  etc. 


Exercise  57 — Oral. 

Do  this  work  by  Section  A  in  the  room  (|  of  class) 
buying  from  Section  B  (other  |  of  class).  Buy,  ask 
questions,  and  both  sections  must  balance  bills. 

rVTII-108) 


PERCENTAGE 

Mrs.  Wilson  bought  a  piano  for  S300.00,  paying 
S50.00  cash  and  agreeing  to  pay  $10.00  per  month  with 
interest  at  6%  per  annum. 

1.  What  is  an  account  of  this  kind  called?     Why? 

2.  What  is  the  unpaid  balance  on  the  principal  of 

this  account  during  the  first  month?     During 
the  last  month? 

3.  How  many  $10.00  payments  must  be  made  on 

this  account?     How  many  months  will  it  take 
to  pay  this  account? 

4.  What  is  the  greatest  amount  on  which  interest 

will  have  to  be  paid  for  1  month?     How  much 
interest  will  be  paid  on  this  sum? 

5.  What  is  the  smallest  amount  on  which  interest 

will  have  to  be  paid  for  1  month?     How  much 
interest  will  be  paid  on  this  sum? 

6.  What  is  the  total  of  the  largest  and  smallest 

interest  payments  on  this  account?    What  is  the 
average  interest  payment? 

7.  How  many  partial  payments  are  to  be  made? 

What  is  the  total  interest  on  this  account? 

8.  What  part  of  this  account  draws  no  interest? 

9.  How  do  we  find  the  interest  on  any  sum  for  1 

month  at  6%  per  annum? 

10.  How  do  we  find  the  largest  interest  pajnnent  to 

be  made?     How  do  we  find  the  smallest  interest 
payment  to  be  made? 

11.  How  do  we  find  the  interest  on  an  installment 

account  most  quickly? 

12.  How  do  we  proceed  when  the  rate  is  5%?     How, 

when  the  rate  is  7%? 

(VIII-109) 


ARITHMETIC 

Exercise  58 — Written. 

Act  out  the  following:   Row  ''A"  buys  of  Row  ''B''; 
''C"of"D'',etc. 

1.  A  furniture  dealer  sold  a  parlor  suite  for  $160.00, 

receiving  $30.00  cash  at  the  time  of  the  sale, 
and  an  agreement  calling  for  $10.00  per  month 
w^ith  interest  at  6% ;  what  is  the  interest  on  this 
account? 

2.  Mr.  Brown  bought  a  house  and  lot  for  $5,500.00, 

paying  $500.00  cash,  and  agreeing  to  pay  $100.00 
per  month  with  6%  interest;  what  is  the  interest 
on  this  account? 

3.  Mrs.  Jackson  bought  a  grand  piano  for  $500.00, 

paying  $50.00  cash  and  agreeing  to  pay  $25.00 
per  month  with  interest  at  6%  per  annum;  what 
is  the  interest  on  this  account? 

4.  A  watch  was  sold  for  $75.00  on  terms  of  $25.00 

cash  and  $5.00  per  month  with  5%  interest; 
w^hat  was  the  amount  of  interest  on  this  account? 

5.  A  kitchen  range  was  sold  for  $60.00  on  terms  of 

$15.00  cash  and  $7.50  per  month  with  interest 
at  7% ;  what  was  the  total  amount  paid  for  the 
range? 

6.  A  building  lot  was  sold  for  $900.00  on  terms  of 

$100.00  cash  and  $80.00  every  second  month 
with  interest  at  6%  per  annum;  what  was  the 
interest  on  this  account? 

7.  A  man  bought  an  automobile  for  $1,200.00  paying 

$200.00  cash  and  agreeing  to  pay  $50.00  per 
month  with  5%  interest;    w^hat  was  the  total 
cost  of  this  automobile? 
(VIII-llO) 


PERCENTAGE 

8.  A  phonograph  costing  $125.00  was  sold  on  terms 

of  $15.00  cash  and  $5.00  per  month  with  5% 
interest;  what  was  the  interest  on  this  account? 

9.  A  building  was  sold  for  $22,000.00  on  terms  of 

$2,000.00  cash  and  $1,000.00  per  year  with  5  J% 
interest;  find  the  total  interest  on  this  sale. 
10.  What  is  the  interest  on  the  sale  of  a  diamond  ring 
for  $135.00  if  $15.00  is  paid  in  cash  at  the  time 
of  the  sale,  and  the  balance  is  paid  at  the  rate  of 
$8.00  per  month  with  7%  interest? 

Lesson  28 

Bank  Discount 

Read  carefully. 

When  banks  loan  money  to  their  depositors  on  prom- 
issory notes,  or  when  they  purchase  notes  signed  by 
third  parties  from  their  depositors,  they  figure  interest 
from  the  date  they  part  with  the  money  to  the  date  of 
the  maturity  of  the  note,  and  deduct  this  interest  from 
the  amount  of  the  note,  giving  the  borrower  the 
remainder. 

The  amount  given  the  borrower  is  called  the  '^  pro- 
ceeds" of  the  note. 

The  amount  retained  by  the  bank  as  interest  is 
called  ''bank  discount." 


EXAMPLE:     Find  the 

proceeds 

of  a  note  for  S300.00 

dated 

March 

1st,   1920, 

due  in 

60  days,  bearing  no     | 

interest 

,  discounted  at  6% 

on  March  1st, 

1920. 

Amount  due  at  the  maturity  of  the  note  =  $300. 

Bank  discount  for  60  davs 

at  6%.. 

=        3. 

Proceeds 

.  .  .  .  =  S297. 

li 

(VIII-111) 


ARITHMETIC 

Since  the  bank  collects  the  full  amount  of  the  note 
at  maturity  and  gives  the  borrower  only  the  proceeds, 
the  difference  between  these  tw^o  amounts  is  the  amount 
of  the  bank  discount. 


EXAMPLE;  Find  the  proceeds  of  a  note  for  $250.  dated  April 
6th,  1921,  due  July  5th,  1921,  bearing  6% 
interest,  discounted  at  5%  on  May  6th,  1921. 

Principal  of  note =  $250.00 

Interest  Apr.  6  to  July  5,  90  days  at  6%  =        3.75 

Amount  due  at  maturity =  $253.75  (Value  of  note) 

Bank  discount  May  6  to  July  5, 

60  days  at  5% =        2.11 

Proceeds =  $251.64 


When  discounting  a  note  which  bears  interest,  the 
interest  due  at  the  maturity  of  the  note  is  added  to  the 
principal,  and  this  entire  amount  is  discounted;  always 
discount  the  full  value  of  the  note. 

Exercise  59 — Oral. 

Play  going  to  a  bank  (some  place  in  the  room)  to 
have  your  note  discounted.  Different  children  may 
take  different  notes;  some  with  interest,  others 
without.    Ask  these  questions.    Be  sure  of  your  money. 

1.  In  the  first  example  shown  in  this  lesson,  what 

amount  is  given  by  the  bank  to  the  borrower? 
What  is  this  amount  called? 

2.  What  amount  must  the  borrower  pay  to  the  bank 

when  the  note  matures?     What  is  this  called? 

3.  Who  gets  the  difference  between  the  amount  given 

by  the  bank  to  the  borrower  and  the  amount 
the  borrower  must  repay  to  the  bank?     Why? 
What  is  this  difference  called? 
(VIII-112) 


PERCENTAGE 

4.  In   the   second   example,   what   amount  will  be 

collected  by  the  bank  when  this  note  matures? 
Of  what  does  this  amount  consist? 

5.  Why  did  the  bank  charge  only  $2.11  for  discount- 

ing this  note,  instead  of  $3.75  which  is  the 
amount  of  interest  to  be  collected  on  the  note? 

6.  When  a  note  for  $200.  bearing  no  interest  is  dis- 

counted for  60  days  at  6%,  what  amount  does 
the  borrower  receive?  What  amount  will  the 
bank  collect  when  the  note  matures?  Who  gets 
the  difference?     Why? 

7.  In  Question  6,  what  is  the  amount  of  the  bank 

discount?    What  is  the  amount  of  the  proceeds? 

8.  On  what  amount  is  bank  discount  computed  when 

a  note  bearing  interest  is  discounted? 

9.  For  what  length  of  time  does  the  bank  compute 

its  bank  discount? 
10.  A  90-day  note  for  $100.  bearing  6%  interest  is 
discounted  by  a  bank  at  5%  60  days  before 
maturity;  state  exactly  how  you  would  find  the 
proceeds  of  this  note. 

Exercise  60 — Written. 

Find  the  proceeds  of  each  of  the  following  notes: 

Rate  j^- 

Date  of  Note        Principal        Time     of       Date  Discounted   -d^?' 

Int. 

I.Apr.   6,1919     $400.      3 mo.   ...   Apr.    6,1919  6% 

2.  May  8, 1920     $350.  120  da.    .  . .   Jul.      7, 1920  5% 

3.  Aug.  4, 1916  $1,500.      4  mo.  6%  Aug.    4, 1916  6% 

4.  Dec.  8, 1923  $280.  90  da.  6%  Jan.  22, 1924  6% 

5.  Oct.  19, 1921  $816.  60  da.  5%  Nov.  18, 1921  6% 

(VIII-113) 


ARITHMETIC 

6.  The  proceeds  of  a  note  are  $394.;   the  bank  dis- 

count at  6%  is  $6. ;  what  is  the  principal  of  the 
note?  For  what  length  of  time  was  the  note 
discounted? 

(Suggestion:     Proceeds  +  Bank    Discount  = 

Principal ; 
$394.  +  $6.  =  $400.; 

$6.  is  the  interest  at  6%  on  $400.  for  how  long? 
Interest  for  1  yr.  =  $24.;    A  =  t  year,  or 
3  mo.) 

7.  The  proceeds  of  a  note  are  $239.00;    the  bank 

discount  for  1  month  is  $1.00;  what  is  the  prin- 
cipal of  the  note?    What  is  the  rate  of  discount? 

8.  I  wish  to  borrow  $500.00  (proceeds)  from  a  bank 

for  4  months  at  6%;  for  what  amount  must  I 
sign  a  note? 

(Suggestion:    4  mo.   interest  at   6%  =  2%; 

therefore,  $500.00  is  98%  of  the  principal. 

If  $500.00  =  98%  of  principal, 

100%o  =  w  of  $500.00,  or 

100%  =  $500.00  -^  .98) 

9.  A  merchant  went  to  his  bank  and  arranged  to 

borrow  $346.50  (proceeds)  for  3  months  at  4%; 
for  what  amount  did  he  sign  a  note? 

10.  The  proceeds  of  a  note  discounted  for  30  days  at 

6%  amounted  to  $358.20;  what  was  the  principal 
of  the  note? 

11.  A  4-months'  note  bearing  6%  interest  was  dis- 

counted 2  months  before  maturity  at  6%,  the 
proceeds  being  $908.82;  what  was  the  principal 

of  the  note? 

(VIII-114) 


PERCENTAGE 


Lesson  29 


Mortgages  and  Bonds 

Read  carefully. 

When  property  is  pledged  as  security  for  the  payment 
of  a  debt,  it  is  done  by  the  signing  of  a  document  called 
a  '' mortgage."  This  mortgage  is  recorded  at  the  office 
of  the  County  Recorder  so  that  any  interested  person 
may  know  that  someone  has  a  temporary  claim  against 
the  property.  When  the  debt  is  settled,  the  original 
mortgage  and  a  form  called  a  '^satisfaction"  are 
obtained  from  the  person  to  whom  the  debt  was  paid, 
and  this  "satisfaction"  is  filed  with  the  County 
Recorder,  after  which  the  property  is  again  free  from 
thie  claim,  encumbrance,  or  lien. 

The  person  who  gives  a  mortgage  is  a  "mortgager." 

The  person  in  whose  favor  a  mortgage  is  given  is  a 
"mortgagee." 

While  a  mortgage  remains  unpaid,  the  fire  insurance 
poHcies  are  left  in  the  hands  of  the  mortgagee  so  that 
he  may  be  protected  in  case  of  fire. 

Mortgages  are  of  two  kinds:  "chattel"  and  "real 
estate."  A  "chattel  mortgage"  pledges  merchandise, 
machinery,  or  other  personal  property  as  security,  while 
a  "real  estate  mortgage"  pledges  houses,  lots,  or  other 
real  property  as  security. 

The  interest  on  a  mortgage  is  payable  at  regular 
stated  intervals,  such  as  once  every  six  months,  once 
every  year,  etc.,  just  as  is  the  interest  on  a  long  time 
promissory  note. 

When  a  large  sum  of  money  is  to  be  borrowed  on  a 
mortgage,    it    is    frequently    done    by    issuing    many 

(VIII-115) 


ARITHMETIC 

''bonds''  for  small  amounts,  the  total  of  which  is  equal 
to  the  amount  to  be  borrowed  on  the  mortgage.  In 
such  cases  the  mortgage  is  made  in  favor  of  a  "trustee" 
who  holds  it  for  the  protection  of  the  bondholders,  as 
each  of  the  bonds  is  really  a  part  of  the  mortgage. 

Most  bonds  have  '^ coupons"  attached,  each  coupon 
representing  the  interest  on  the  bond  for  a  quarterly 
or  semi-annual  period;  each  of  these  coupons  is  dated 
ahead  to  correspond  with  the  date  on  which  it  falls 
due,  and  when  that  date  arrives,  it  must  be  clipped  off 
the  bond  and  cashed  through  a  bank  just  as  if  it  were 
a  check  for  so  much  interest. 

Government  bonds  are  similar  to  other  bonds,  but, 
of  course,  they  need  no  mortgage  to  protect  them,  as 
they  are  an  obligation  of  the  government  itself,  and  it 
is  beyond  question  that  they  will  be  paid  at  maturity. 

Never  confuse  '^bonds''  with  '^ stocks,"  for  they  are 
as  different  as  day  and  night.  Stocks  will  be  considered 
in  the  next  lesson. 

Exercise  61 — Oral. 

Act  these  out  in  the  class  room.  Appoint  a  recorder. 
Watch  the  papers  handled.     Talk  freely. 

1.  Frank  Graham  borrowed  $1,000.00  on  March  5, 
1920,  on  his  house  and  lot,  from  William  Rich, 
for  3  years  at  6%  payable  semi-annually. 
(a)  What  kind  of  a  mortgage  was  Mr.  Graham 
required  to  sign?    Who  was  the  mortgager? 
Who  was  the  mortgagee? 
(6)  What  did  Mr.  Rich  do  with  this  mortgage  so 
/  that  any  interested  person  might  find  out 
(VIII-116) 


PERCENTAGE 

that  he  had  a  claim  or  lien  against  the 
property?  Who  holds  the  fire  insurance 
policies  until  this  mortgage  is  paid? 

(c)  When  was  the  first  interest  payment  due  on 

this  mortgage?  What  amount  of  interest 
was  due  on  that  date?  When  was  the  second 
interest  payment  due?  What  amount? 
Name  the  dates  on  which  each  of  the  other 
interest  payments  were  due. 

(d)  When  does  this  mortgage  fall  due?      What 

amount  must  be  paid  on  that  date? 

(e)  When  this  mortgage  was  paid  in  full,  what  two 

documents  did  the  mortgager  demand  from 
the  mortgagee?  What  did  the  mortgager 
do  to  make  it  known  that  this  lien  on  the 
property  had  been  satisfied? 

2.  When  machinery  is  pledged  to  secure  payment  of 

a  debt,  what  kind  of  a  mortgage  is  given? 

3.  Henry  Walker,  a  printer,  borrowed  $400.00  on  his 

printing  presses  from  the  Merchants  Loan  Com- 
pany by  giving  a  mortgage  dated  Oct.  4,  1920, 
due  in  one  year,  bearing  7%  interest  payable 
quarterly. 

(a)  What  kind  of  a  mortgage  was  given  in  this 

transaction?  AVho  was  the  mortgager? 
Who  was  the  mortgagee? 

(b)  State  what  the  Merchants  Loan  Company  did 

to  protect  its  interests  when  this  mortgage 
was  received.  Who  has  possession  of  the 
fire  insurance  policies  while  this  mortgage 
remains  unpaid? 

(\^III-ll7) 


ARITHMETIC 

(c)  State  the  dates  on  which  the  interest  must  be 

paid.      State  the  amount  of  each  interest 
payment. 

(d)  When  does  this  mortgage  fall  due?     What 

amount  must  be  paid  on  that  date? 

(e)  State  what  Henry  Walker  did  to  protect  his 

interest  after  paying  his  mortgage. 
4.  The  Empire  Manufacturing  Company,  wishing  to 
raise  $500,000.00  to  enlarge  its  manufacturing 
facihties,  issues  a  series  of  one  thousand  $500.00 
bonds  dated  July  1,  1919,  protected  by  deposit 
with  the  First  Trust  Co.,  as  Trustee,  of  a  mort- 
gage on  all  its  buildings.  The  bonds  bear  6% 
interest  payable  Jan.  1st  and  July  1st  of  each 
year  and  mature  as  follows: 

i  or  $100,000.  July  1st,  1924. 
i  or  100,000.  July  1st,  1925. 
i  or  100,000.  July  1st,  1926. 
i  or  100,000.  July  1st,  1927. 
i  or    100,000.  July  1st,  1928. 

(a)  Who  is  the  mortgager?  Who  is  the  mortgagee? 
What  kind  of  a  mortgage  was  given? 

(6)  What  did  the  First  Trust  Co.  do  for  the  pro- 
tection of  the  bondholders  when  this  mort- 
gage was  received?  What  was  done  with 
the  fire  insurance  policies? 

(c)  State  how  much  interest  must  be  paid  on  each 

$500.00  bond  on  Jan.  1st,  1920.    How  much 
on  July  1st  of  every  year? 

(d)  State  how  the  bondholders  will  collect  their 

interest  on  Jan.  1st  and  July  1st  of  each  year. 
(VIII-118) 


PERCENTAGE 

.  {e)   On  what  date  would  this  mortgage  be  fully 

I  satisfied? 

(/)  What  would  the  Empire  Manufacturing  Com- 
pany do  to  protect  its  interests  after  fully 
satisfying  this  mortgage? 
f        5.  The  interest  on  the  Fourth  Liberty  Loan  issued  by 
the  United  States  in  1918  to  help  finance  the 
World  War  is  payable  semi-annually  at  4J%; 
for  what  amount  is  each  semi-annual  interest 
coupon  on  a  $100.00  bond? 

6.  How  do  you  collect  the  interest  on  such  a  bond? 

7.  Is  this  issue  of  bonds  protected  by  a  mortgage? 
If  not,  what  assurance  has  a  bondholder  that  the 
bond  will  be  paid  at  maturity? 

Exercise  62 — Written. 

1.  A  mortgage  for  $9,500.00  dated  Aug.  1st,  1918, 
bearing  5%  interest  payable  semi-annually, 
matures  in  5  years.  What  is  the  total  amount 
of  interest  which  would  be  paid  on  this  mortgage? 

2.  A  3-year  mortgage  for  $8,700.00  bearing  6%  inter- 
est matures  Oct.  1,  1923,  on  which  date  $4,700. 
and  the  interest  to  date  is  paid  and  the  balance 
is  renewed  for  another  3-year  period.     W^hat  is 

'the  total  interest  on  this  mortgage? 

3.  A  house  and  lot  worth  $8,000.00  is  mortgaged  for 
60%  cf  its  value  for  5  years  at  5|%;  what  is  the 
total  amount  of  interest  on  this  mortgage? 

4.  A  property  owner,  wishing  to  borrow  the  largest 
amount  on  which  he  can,  without  inconvenience, 
pay  6%  interest,  figures  that  he  can  spare  $261.00 

(VIII-119) 


ARITHMETIC 

a  year  to  meet  the  interest  payments.  What 
amount  can  he  borrow? 

5,  A  certain  bond  issue  is  dated  Dec.  1,  1920,  and 

consists  of  four  hundred  $500.00  bonds  bearing 
6%  interest  payable  semi-annually  on  June  1st 
and  Dec.  1st  of  each  year.  The  bonds  are  in 
three  series: 

Series  ''A"  is  for  20%  of  the  total,  and  expires 

Dec.  1,  1925. 
Series  ^'B"  is  for  30%  of  the  total,  and  expires 

Dec.  1,  1926. 
Series  '^C  "  is  for  50%  of  the  total,  and  expires 
Dec.  1,  1927. 
(a)  What  sum  must  be  paid  on  the  principal  on 
each  of  these  dates:    Dec.  1,  1925;    Dec.  1, 
1926;  Dec.  1,  1927? 
(6)  WTiat  amount  of  interest  must  be  paid  on  each 
of  these  dates:  June  1,  1921;  Dec.  1,  1925; 
June  1,  1926;  Dec.  1,  1926;  Dec.  1,  1927? 
(c)   What  is  the  total  interest  on  this  issue  of  bonds? 

6.  The  interest  coupons  on  a  $500.00  6%  bond  are 

payable  Jan.  1st  and  July  1st  of  each  year.  If 
I  bought  this  bond  on  April  1st  how  much 
accrued  interest  would  I  have  to  pay?  How  and 
when  would  I  get  this  accrued  interest  back? 

Lesson  30 
Corporations  and  Their  Capital  Stock 

Read  all  of  this  very  carefully. 

When  a  business  is  owned  by  one  person  or  by  a 
few  individuals  in  partnership,  the  capital  needed  for 

(VIII-120) 


PERCENTAGE 

its  operation  is  naturally  limited  to  the  financial  re- 
sources of  the  one  person  or  few  people  who  own  it. 

As  most  of  the  larger  commercial  enterprises  of  the 
present  day  operate  on  so  great  a  scale  that  practically 
no  one  person  and  but  very  few  small  groups  of  persons 
could  possibly  command  the  vast  sums  of  money  which 
are  needed  for  their  operation,  it  was  necessary  to  devise 
a  method  whereby  many  persons  might  invest  their 
money  in  such  enterprises  without  necessarily  devoting 
their  time  to  the  management  of  the  business.  To 
meet  this  necessity,  the  modern  ^'corporation^'  was 
evolved. 

A  '^  corporation '^  is  an  artificial  body  created  and 
chartered  by  the  law  of  the  state  to  transact  its  specific 
business  as  if  it  were  an  individual.  Can  you  give  the 
name  of  a  corporation  doing  business  in  the  United 
States? 

The  capital  of  a  corporation  is  limited  to  a  specified 
amount  by  its  charter,  and  this  specified  amount  is 
divided  into  a  certain  number  of  shares,  each  of  which 
has  a  like  ''par"  or  face  value.  Thus,  if  a  corporation  is 
capitalized  for  $100,000.  and  this  is  divided  into  1,000 
shares,  then  each  share  has  a  par  value  of  $100. 

The  shares  are  represented  by  ''stock  certificates" 
which  show  the  par  value  of  each  share,  the  number  of 
shares  owned,  and  the  name  of  the  owner;  therefore, 
a  stock  certificate  of  a  corporation  is  proof  that  the 
owner  is  a  partner  in  the  business  to  the  extent  that 
the  ratio  of  the  number  of  shares  held  by  him  bears  to 
the  total  mmaber  of  shares  of  the  corporation. 

The  affairs  of  a  corporation  are  managed  by  a  "board 
of  directors"  who  are  elected  by  the  stockholders.     Id 

(VIII-121) 


ARITHMETIC 

voting  for  the  directors,  each  share  of  stock  usually 
entitles  the  owner  to  one  vote  for  each  director  to  be 
elected;  thus  it  is  readily  seen  that  the  owner  of  a  few 
shares  of  stock  has  very  little  to  say  regarding  the 
management  of  the  business.  The  directors  elect  the 
'^officers"  who  sign  the  official  papers  of  the  corpora- 
tion, such  as  checks,  notes,  etc. 

The  profits  of  a  corporation  are  distributed  to  the 
stockholders  in  the  form  of  '^dividends''  which  are  paid 
at  such  time  and  in  such  amounts  as  the  directors  may 
decide;  therefore,  the  income  from  shares  of  stock 
depends  entirely  upon  the  profits  of  the  corporation  and 
is  not  a  debt  which  must  be  paid,  as  is  the  interest  on 
bonds.  Furthermore,  while  the  bondholders  are  usually 
protected  by  a  mortgage  and  are  therefore  the  first  to 
receive  their  money  if  the  corporation,  for  any  reason, 
goes  out  of  business,  the  stockholders  are  the  owners  of 
the  business  and  are,  therefore,  the  last  to  receive  their 
money — if  there  is  any  left  after  paying  all  the  other 
claims,  and  those  stockholders  who  did  not  pay  the 
full  face  or  par  value  for  their  stock  are  liable  to  be 
called  on  to  pay  the  difference  between  what  they  have 
paid  and  the  par  value,  and  if  the  corporation  con- 
ducts a  bank,  the  stockholders  are  liable  to  pay  double 
the  par  value  of  their  stock  so  that  the  depositors  may 
suffer  no  loss.  Thus,  we  say  ^gain:  Do  not  confuse 
bonds  with  stocks. 

Exercise  63 — Oral. 

1.  The  authorized  capital  stock  of  a  corporation  is 
5,000  shares  of  a  par  value  of  $100.  each;  what 
is  the  total  capitalization  of  this  corporation? 

(VIII-122) 


PERCENTAGE 

What  is  the  par  value  of  20  shares  of  this  stock? 
What  is  the  par  value  of  100  shares? 

2.  What  percentage  of  this  business  belongs  to  a 

man  who  owns  500  shares  of  stock?  2,500 
shares?     4,000  shares?     5,000  shares? 

3.  If  this  corporation  pays  an  annual  dividend  of  8%, 

what  would  be  the  total  amount  of  such  an 
annual  dividend?  How  much  would  a  person 
who  owns  1  share  receive?  How  much  would 
the  owner  of  20  shares  receive?  100  shares? 
1,000  shares? 

4.  Who  elects  the  directors  of  a  corporation?     Who 

elects  the  officers?  Has  the  owner  of  a  few 
shares  of  stock  very  much  to  say  regarding  the 
management  of  the  business? 

5.  A  corporation  is  authorized  by  its  charter  to  manu- 

facture machinery;  can  this  corporation  go  into 
the  banking  business?  Into  the  railroad  busi- 
ness?    Explain  why  you  answer  as  you  do. 

6.  A  corporation,  in  accordance  with  its  charter,  has 

an  authorized  capitalization  of  $100,000.  con- 
sisting of  1,000  shares  of  a  par  value  of  $100. 
each;  can  this  corporation  sell  1,500  shares? 
Explain  why? 

7.  By  what  authority  is  a  corporation  permitted  to 

organize  and  exist? 

8.  Is  a  stockholder  a  creditor  of  a  corporation?     Is  a 

bondholder  a  creditor?     Explain  fully. 

9.  Are  the  dividends  on  shares  of  stock  a  debt  which 

must  be  paid  regardless  of  the  profits  of  a  cor- 
poration? How  about  the  interest  on  bonds? 
Explain  fully. 

(VIII-123) 


ARITHMETIC 

10.  A  corporation  goes  out  of  business  and  its  property 

is  sold  for  $175,000.00;  the  bondholders'  claims 
amount  to  $50,000.00;  ordinary  creditors'  claims 
amount  to  $50,000.00;  stockholders'  claims 
amount  to  $100,000.00;  in  what  order  are  these 
claims  paid?     Who  are  the  losers?     Why? 

11.  Supposing  the  stockholders  in  Question  10  had 

bought  their  stock  at  $50.00  per  share,  for  shares 
of  a  par  value  of  $100.00,  how  much  might  the 
holder  of  each  share  be  called  on  to  pay  in  ad- 
dition to  the  $50.00  he  has  paid?    Explain  fully. 

12.  Supposing  the  corporation  in  Question  10  were  en- 

gaged in  the  banking  business,  and  the  stock- 
holders have  bought  their  shares  for  $50.00  per 
share  for  each  $100.00  share,  how  much  might 
the  holder  of  each  share  be  called  on  to  pay  in 
addition  to  the  $50.00  he  has  paid?  Explain  fully. 

Exercise  64 — Written. 

1.  The  net  profits  of  a  corporation  for  a  certain  year 

are  $15,000.;  the  capital  stock  is  $200,000.; 
what  per  cent  of  dividends  may  be  declared  by 
the  directors? 

2.  What  per  cent  of  dividends  may  the  directors 

declare  if  the  net  earnings  are  $4,375.  and  the 
capital  stock  is  $87,500.? 

3.  The  net  profits  of  a  corporation  capitalized  at 

$100,000.  enabled  the  directors  to  declare  a 
dividend  of  8%  and  set  aside  the  remainder 
amounting  to  $15,000.  to  be  used  as  dividends 
in  future  years;  what  were  the  net  profits? 

(VIII-124) 


PERCENTAGE 

4.  The  directors  of  a  corporation  declared  a  5% 
dividend  amounting  to  $50,000.;  what  was  the 
amount  of  the  capital  stock  of  this  corporation? 

5.  The  capital  stock  of  a  corporation  is  $300,000.  and 
the  par  value  of  each  share  is  $100.;  this  com- 
pany pays  a  semi-annual  dividend  of  4% ;  what 
must  its  annual  net  profits  amount  to? 

6.  What  would  be  the  annual  income  of  a  man  who 
owns  5,500  shares  of  stock  of  a  par  value  of  $100. 
each,  if  the  corporation  pays   1|%  dividends 

quarterly? 

7.  The  net  profits  of  a  corporation  are  sufficient  to 

pay  dividends  of  2|%  quarterly  on  a  capitaliza- 
tion of  $150,000.  after  settmg  aside  $10,000.  for 
emergencies  and  allowing  $3,500.  to  remam  un- 
distributed. What  per  cent  of  dividends  might 
the  directors  have  declared  if  they  had  chosen 
to  do  so? 

8.  What  per  cent  of  dividends  might  the  directors 

of  a  corporation  declare  quarterly  if  the  total 
annual  earnings  of  the  business  are  $75,000.,  the 

r  total   annual   expenses   are   $45,000.,    and   the 

capital  stock  is  $300,000.? 
^         9.  A  stockholder  received  $770.00  as  his  7%  dividend 
r  on  22  shares  of  stock;   what  was  the  par  value 

of  this  stock? 
10.  A  stockholder  received  $3,300.  as  his  6%  dividend 
on  4:5tt%  of  the  stock  of  a  corporation;   what 
I  was  the  entire  capital  stock?     If  the  par  value 

of  the  shares  is  $1,000.,  how  many  shares  did 
this  stockholder  own? 


ARITHMETIC 


Exercise  65 — Oral  and  Written. 

Problem  Project. 

Organize  a  corporation — all  the  children  must  be 
stockholders.  (Build  your  corporation  on  what  you 
know.) 

1.  Study  the  purpose  of  your  L^,^   ^      '  .,    , 

, .  i  What  can  it  do,  and 

corporation.  u  v  ^  9 

t  what  can  t  it  do? 

2.  Capital  needed;  how  much? 
/  Who  elects  them?     How? 
\  What  are  their  duties? 

Who  elects  them?     How? 
What  are  their  duties? 

'  Fixed  rate  of  divi- 
(a)  Preferred  1  dends,  paid  first  out 


3.  Directors 

4.  Officers 


5.  Issue  capital 
stock  certif-  < 
icates. 


(b)  Common 


6.  Buy    and     sell 
capital  stock. 


N.  B. 


[  of  profits. 
Unlimited    rate    of 
dividends,  paid  after 
preferred  dividends 
,  are  paid. 
^  (a)  At  par  value. 

(b)  At  market  value. 

(c)  Brokerage. 

r  Rate  indicated. 

(d)  Income  i  Calculate  your 

,  dividends. 


Art  Work:  Hang  posters  to  boom  your  project. 

f 

Enghsh:  Write  letters  < 


,  questions  about 

and        I  .    ^ 

your  project. 
L  answering  J 

(VIII-126) 


PERCENTAGE 

Lesson  31 

Rate  of  Income  (Yield)  on  Stocks  and  Bonds  Bought 
at  a  Premium  or  Discount 

Read  carefully. 

Many  of  the  larger  corporations  have  two  kinds  of 
capital  stock,  '^preferred  stock"  and  ''common  stock." 

''Preferred  stock"  is  stock  on  which  dividends  at  a 
certain  fixed  rate  per  cent  must  be  paid  before  any  other 
dividends  may  be  paid. 

"Common  stock"  is  stock  on  which  unlimited  divi- 
dends may  be  paid  after  preferred  dividends  have  been 
provided  for. 

Thus,  where  the  net  earnings  are  only  slightly  in 
excess  of  the  preferred  dividends,  the  preferred  stock  is 
the  more  valuable  because  the  preferred  dividends  must 
be  paid  in  full  before  any  dividends  can  be  paid  on  the 
common  stock;  but  where  the  net  earnings  are  greatly 
in  excess  of  the  preferred  dividends,  the  common  stock 
is  the  more  valuable  because  the  dividends  thereon 
might,  under  certain  circumstances,  be  many  times 
greater  than  on  the  preferred  stock. 

Bonds,  of  course,  always  specify  the  rate  of  interest 
that  they  bear. 

The  "par"  value  of  stock  is  the  value  per  share  which 
is  indicated  on  the  certificate  of  stock. 

The  "market"  value  of  stock  is  the  value  at  which 
the  shares  can  be  sold  in  the  market  at  a  given  time. 

If  the  market  value  is  greater  than  the  par  value, 
the  stock  is  said  to  be  "above  par"  or  ^'at  a  premium." 

If  the  market  value  is  less  than  the  par  value,  the 
stock  is  said  to  be  "below  par"  or  "at  a  discount." 

CV7II-127) 


ARITHMETIC 

All  that  has  been  said  regarding  the  par  value  and 
the  market  value  of  stocks  may  also  be  said  about 
bonds. 

Bonds  and  stocks  are  bought  and  sold  through 
brokers  who  receive  a  brokerage  of  about  |%  on  the 
par  value  from  the  buyer  and  also  from  the  seller  of 
the  bonds  or  stocks.  Thus,  when  stock  is  bought  at 
98,  a  SIOO.  share  costs  the  buyer  $98.  +  Sf  brokerage 
=  S98.12J,  but  the  seller  receives  only  S98.  -  $| 
brokerage  =  $97.87^;  therefore,  the  broker  receives 
12  §^  from  the  buyer  and  12J^  from  the  seller,  or  25^ 
in  all. 


EXAMPLE  #1:     Find  the  yield  on  ten  $100.  shares  of  6%  stock 
bought  at  95. 

Cost:  10  shares  @  $95.  +  $1 =  $951.25 

Dividend  on  $1,000  @  6% =      60.00 

.00  annual  dividends  on  an  investment  of  $951.25  =  /^rfA 
of  100%;  or  ^%-\%%  of  100%;  or  Qj%k%,  Ans. 


Dividends  are  always  paid  on  the  par  value  of  the 
stock,  but  to  find  the  rate  per  cent  of  income  on  money 
invested  in  stocks,  we  must  consider  the  actual  cost 
(including  brokerage)  and  not  the  par  value,  and  we 
must  assume  that  the  market  value  of  the  stock  will 
remain  unchanged  because  there  is  no  way  of  deter- 
mining the  price  for  which  the  stock  can  be  sold  at 
some  future  time  until  that  time  actually  arrives. 

After  the  stock  has  been  sold  and  we  have  deter- 
mined our  loss  or  gain  on  the  transaction,  such  loss 
or  gain  must  be  taken  into  consideration  as  shown  in 
Examples  #4  and  #5. 

(VIII-128) 


PERCENTAGE 


EXAMPLE  #2:      Find  the  yield  on  a  5-year  6%  $100.  bond 

bought  at  94|. 
Par  value  payable  in  5  years =  $100.00 1    Average  in  vest- 
Cost  $94 1  +  $1 =       95.00  /     ment  $97.50. 

Discount  gained  during  life  of  bond  =       $5-00  =  $1.00  per  year 

Annual  interest  to  be  received  (6%  on  $100.00) =  $6.00 

Plus  one  year's  portion  of  discount =     1.00 

Total  annual  earnings =  $7.00 

$7.00  annual  earnings  on  an  average  investment  of  $97.50  = 

7x50 %>  Ans. 

.0718,  Rate. 

97.50)7.000000 

EXAMPLE  #3:     Find  the  yield  on  a  10-year  7%  $1,000.  bond 
bought  at  109|. 

Cost  $1,098.75  +  $1.25 =  $1,100,001  Average  invest- 

Par  value  payable  in  10  years =     1,000.00]      ment  $1,050. 

Premium  lost  during  life  of  bond  =      $100.00  =  $10.  per  yr. 

Annual  interest  to  be  received  (7%  on  $1,000.) =  $70.00 

Minus  one  year's  portion  of  premium =     10.00 

Net  annual  earnings =  $60.00 

00  annual  earnings  on  an  average  investment  of  $1,050.  = 
5tVo%,  Ans. 


Bonds  are  always  due  and  payable  at  their  par  value 
on  a  definitely  known  date;  therefore,  any  premium 
paid  for  the  bond  is,  in  fact,  a  sacrifice  of  a  portion  of 
the  interest  to  be  collected  during  the  life  of  the  bond, 
while  any  amount  that  may  be  saved  by  buying  the 
bond  at  a  discount  is  really  an  addition  to  the  interest 
to  be  collected  during  the  life  of  the  bond.  Such 
premium  or  discount  must  be  distributed  over  the  life 
of  the  bond  and  only  one  year's  portion  thereof  con- 
sidered when  finding  the  rate  of  income,  and  we  must 
figure  the  rate  of  income  on  the  average  value  of  the 
bond;  that  is,  the  value  which  is  half  way  between 
the  cost  price  and  the  par  value. 

(VIII-129) 


ARITHMETIC 


EXAMPLE  #4:  Find  the  yield  on  a  $100.00  share  of  stock 
paying  6%  annual  dividends,  bought  at 
93|  and  sold  3  years  later  at  97|. 

Sold  for  $97|  -  $| =  $97,001  Average  investment 

Cost  $931  +  $1 =    94.00/  $95.50. 

Increase  in  value  during  3  years. .  =    $3.00  =  $1.00  per  yr. 

Annual  dividends  (6%  on  $100.00) =  $6.00 

Plus  one  year's  portion  of  increase  in  value =    100 

Total  annual  earnings =  $7.00 

$7.00  annual  earnings  on  an  average  investment  of    $95.50  = 
7t'A%,  Ans. 

EXAMPLE  #5:  Fmd  the  yield  on  a  $100.00  share  of  stock 
paying  8%  annual  dividends,  bought  at 
102 1  and  sold  2  years  later  at  99|. 

Cost  $1021  +  $1 =  $103,001  Average  investment 

Sold  for  $991  -  $1 =      99.00/  $101.00. 

Decrease  in  value  during  2  years  =      $4.00  =  $2.00  per  year. 

Annual  dividends  (8%  on  $100.00) =  $8.00 

Minus  one  year's  portion  of  decrease  in  value =    2.00 

Net  annual  earnings =  $6.00 

$6.00  annual  earnings  on  an  average  investment  of    $101.00  = 
5t«A%.  Ans. 


When  stock  is  actually  sold  for  more  or  less  than 
was  paid  for  it,  the  gain  or  loss  must  be  distributed 
over  the  period  during  which  the  stock  was  owned,  and 
the  yield  must  be  based  on  the  average  value  of  the 
stock  in  the  same  manner  as  we  figure  the  yield  on 
bonds  which  were  bought  at  a  premium  or  discount. 

Exercise  66 — Oral. 

Use  the  following  in  asking  your  chosen  corporation 
officials  for  information.     They  must  be  sure  to  know. 
1.  What  is  the  difference  between  preferred  stock  and 
common  stock  as  you  understand  it? 

(viii-ino) 


PERCENTAGE 

2.  If  a  corporation  has  both  preferred  and  common 

stock  outstanding  and  its  net  earnings  are  only 
slightly  in  excess  of  the  amount  required  for  the 
dividends  on  the  preferred  stock,  which  class  of 
stock  would  be  the  more  valuable?     Why? 

3.  In  Question  2,  which  would  be  the  more  valuable 

if  the  net  earnings  were  greatly  in  excess  of  the 
preferred  dividends?     'WTiy? 

4.  What  is  the  par  value  of  a  share  of  stock  or  of  a 

bond? 

5.  What  is  the  market  value  of  a  share  of  stock  or  of 

a  bond? 

6.  When  is  a  bond  sold  at  a  premium?    At  a  discount? 

7.  How  do  stock  brokers  charge  their  brokerage  on 

sales  of  stocks  or  bonds? 

8.  How  would  you  find  the  rate  of  income  (yield)  on 

a  $100.00  share  of  7%  stock  bought  at  104i? 

9.  If  a  $1,000,  5%  bond,  bought  at  94 J,  matures  in 

5  years,  what  amount  will  be  collected  at 
maturity?  How  much  discount  is  earned  on 
this  bond?  Is  this  discount  earned  in  one  3^ear 
or  during  the  life  of  the  bond?  What  should  be 
done  with  the  discount  so  that  the  correct  per- 
centage of  annual  income  may  be  ascertained? 
What  is  the  average  investment  in  this  case? 

10.  How  would  you  find  the  yield  on   the  bond  in 

Question  9? 

11.  If  a  $100.00,  7%  bond,  bought  at  1091,  matures 

in  10  years,  was  the  bond  bought  at  a  premium 
or  at  a  discount?  How  much  premium  or  dis- 
count?    '^^Tiat  is  the  average  investment  in  this 


case? 


(VlII-131) 


ARITHMETIC 

12.  How  would  you  find   the  yield  on  the  bond  in 

Question  11? 

13.  The  following  list  shows  the  sales  (for  a  week)  of 

some  of  the  securities  on  the  Chicago  Stock 
Exchange : 

Stocks 

Com'pany  Sales  High    Low      Close     from  Previous 

Week 

Am.  Radiator 6  298    295    298  -  2 

doPfd 2  116     116     116  -6 

Am.  Shipbuilding 967  119^  108^  119|  +1| 

Quaker  Oats 5  260     260     260  +5 

doPfd 11     99i    99      99 

Sears  Roebuck 906  163     160     163  +3 

Bonds 

C.  C.  Rys 5%  $1,000.  61  61  61 

C.  Rys 5%  $1,000.  58i  58i  58i        - 1 

Chicago  Tel.  Co.,  5%  $1,000.  96i  96i  96^ 

Swift  &  Co 5% $7,500.  97  96f  96f        -^ 

Name  the  items  which  closed  above  par.      Name 
those  which  closed  below  par. 

14.  State  a  possible  reason  why  Am.  Radiator  com- 

mon stock  should  be  quoted  at  298  while  the 
same  company's  preferred  stock  is  quoted  at 
only  116? 

15.  Which  stock  would  be  the  better  investment  as 

regards  yield,  Am.  Radiator  at  298  or  Quaker 
Oats  at  260,  if  the  dividend  rates  of  the  two 
stocks  are  alike? 

(VIII-132) 


PERCENTAGE 

Exercise  67^ — Written. 

(All  stocks  in  the  following  examples  are  to  be  con- 
sidered as  of  $100.00  par  value  and  brokerage  is  to  be 
computed  at  |%  unless  otherwise  stated.) 

1.  What  is  the  cost  of  45  shares  of  U.  S.  Iron  Co. 

common  stock  at  98  f? 

2.  I  sold  60  shares  of  Am.  Wool  Co.  common  stock 

at  58 1  and  invested  the  proceeds  in  Am.  Wool 
Co.  preferred  stock  at  86|;  how  many  shares 
of  preferred  stock  did  I  receive? 

3.  A  certain   common   stock  pays   12%   dividends 

annually;  what  would  be  the  per  cent  of  income 
on  1  share  of  this  stock  bought  at  111  J?  What 
per  cent  on  15  shares? 

4.  A  stock  called  '^Consohdated  Railway  Corpora- 

tion 7%  Preferred"  sells  on  a  basis  which  nets 
the  investor  10 J%;  what  is  this  stock  quoted 
on  the  stock  exchange? 

5.  A  packing  company's  8%  stock  is  selling  on  a 

basis  which  nets  the  investor  6%;  how  many 
shares  of  this  stock  must  be  bought  to  insure 
an  annual  income  of  $720.00?  What  is  this 
stock  quoted  on  the  stock  exchange?  ^Tiat 
amount  must  be  invested? 

6.  What  is  the  per  cent  of  income  or  yield  on  a 

6-year  5%  $100.00  bond  bought  at  931? 

7.  What  is  the  per  cent  of  yield  on  eight  10-year  6% 

$100.00  bonds  bought  at  109i? 

8.  What  is  the  yield  on  a  50-year  5%  $1,000.  bond 

bought  at  891? 

9.  What  is   the   difference   in   the  rate   of  income 

(VIII-133) 


ARITHMETIC 

between  these  two  investments,  and  which  is  the 

more  profitable: 

A  share  of  7%  stock  bought  at  109J,  or 
A  6-year  6%  $100.00  bond  bought  at  98|? 

10.  A  20-year  6%  $500.00  bond  bought  at  951  yields 

what  per  cent  of  income? 

11.  Find  the  yield  on  a  $100.00  share  of  stock  pajdng 

7%  annual  dividends  bought  at  941-  and  sold 
6  years  later  at  101  J. 

,  Lesson  32 

Insurance 
Read  carefully. 

^^ Insurance"  is  a  sum  of  money  promised  to  be  paid 
by  an  Insurance  Company  to  the  insured  person  or 
company  in  case  of  loss  or  injury  of  a  certain  kind. 

The  written  contract  given  the  insured  is  the 
^^  policy.'' 

The  amount  of  money  promised  in  the  policy  is  the 
^'face"  of  the  policy. 

The  amount  charged  for  insurance  is  the  '^premium." 
There  are  many  kinds  of  insurance,  but  those  most 
commonly  used  are : 

Life  Insurance,  which  is  insurance  against  the  loss 

of  life. 
Accident  Insurance,  which  is  insurance  against  dis- 
ability on  account  of  accident. 
Fire  Insurance,  which  is  insurance  against  the  loss 

or  injury  of  property  by  fire. 
Marine  Insurance,  which  is  insurance  against  the 
loss  of  property  at  sea. 

(VIII-134) 


PERCENTAGE 

Employers'  Liability  Insurance,  which  is  insurance 
against  loss  by  reason  of  claims  for  damages  by 
workmen  on  account  of  injury  they  may  sustain 
w^hile  employed/ 
Fidelity  Insurance,  which  is  insurance  against  loss 
on  account   of  the  dishonesty   of  persons  oc- 
cupying positions  of  trust. 
Most  insurance  premiums  are  stated  at  a  certain 
price  per   $100.00   of   insurance,   but   frequently   the 
premium  is  stated  at  a  certain  rate  per  cent.     The  rate 
of   premium,    of   course,    depends   entirely   upon   the 
hazards  connected  with  the  risk;    thus,  fire  insurance 
rates  on  a  wooden  building  are  usually  much  higher 
than  on  a  stone  building,  etc. 

Exercise  68 — Oral. 

Children  must  be  ready  to  ask  and  answer  these  when 
class  room  becomes  an  Insurance  Office;  so  be  careful. 

The  National  Insurance  Company  insures  William 
Burns'  furniture  against  loss  by  fire  to  the  extent  of 
$1,000.  at  80^  per  year  per  $100.00: 

1.  ^^Tio  is  the  insured?    Who  is  the  insurer? 

2.  What  is  this  written  agreement  called? 

3.  What  name  is  given  to  the  amount  paid  to  the 

insurance    company    for    assuming    this    risk? 
How  much  is  this  amount  on  this  policy? 

4.  Wliat  is  the  face  of  this  policy? 
.  5.  What  kind  of  insurance  is  this? 

6.  Examine  an  insurance  policy  and  tell  us  what  3^ou 

can  about  it. 

7.  Mention  and  explain  some  other  kinds  of  insurance. 

(VIII-135) 


ARITHMETIC 

8.  What  is  life  insurance?     Accident  insurance? 

9.  What  is  marine  insurance?     Employers'  liability 

insurance? 

10.  What  is  fidelity  insurance? 

11.  If  the  premium  is  stated  at  a  certain  price  per 

$100.00  of  insurance,  how  do  we  find  the  entire 
premium  on  a  pohcy? 

12.  If  the  premium  is  stated  at  a  certain  rate  per  cent, 

how  do  we  find  the  entire  premium  on  a  policy? 

13.  On  what  does  the  rate  of  premium  chiefly  depend? 

14.  Where  should  we  keep  insurance  policies? 

15.  Mention  several  kinds  of  insurance  that  every 

owner  of  an  automobile  should  have. 

16.  What  kind  of  insurance  should  a  landlord  have? 

Exercise  69 — Oral  and  Written. 

Problem  Project 

Insurance 

Make  your  Classroom  an  Insurance  Office 

(a)  Marine 
(6)   Fire 
.  .       .„        J  1.  Straight 

(c)  Lite       j  2.  Endowment 

(d)  Employers'  Liability 

(e)  Fidehty 
(/)  Accident 


1.  Different  desks  for 
difi'erent  depart- 
ments. 


^    _.     ^        .  .        ,    'Pay  Premiums  Here" 

ous'^'lacer"'^''"''' ^  "  Fire  Insurance  " 

^       ^*  ['•  Accident  Insurance,  etc. ^ 

(From  Art  Dept.) 

(VIII-136) 


PERCENTAGE 


3.  Get  ready  to  question 
officials  preparatory  to 
taking  out  insurance. 
Seek  to  know  all  about 
it  now. 


This  can  be  done  partly 
by  writing  letters  ask- 
ing for  information,  and 
answering  them ;  or  by 
reading  some  received. 


(Correct  English  is  essential.) 
4.  Take  out  insurance 


Rate  and  premium  paid. 

„,  Does  policy  cover  correctly? 

now;  watch:  ]  Privileges  granteds^ 

[  What  is  doneV\^ctbg»licy. 
(Thoroughness  in  Topic   3  will78>[K\N(D)iow.) 

5.  Collecting  on  a  policy  f  After  fire.  ^  ^ 

(a)  Adjuster  -{  After  accident. 

(b)  Signing  of  papers    [  After  maturity. 

Exercise  70 — Written. 

Play  your  part  in  any  of  these  in  your  Insurance 
Office. 

1.  A  farmer  valued  his  barn  at  $900.,  his  house  at 

$2,400.,  his  farm  implements  at  $1,200.  and  his 
furniture  at  $750. ;  he  insured  all  of  these  items 
at  85%  of  their  value  at  i^% ;  what  was  the 
premium  on  this  policy? 

2.  A  certain  life  insurance  policy  stipulates  that  if 

the  premium  is  paid  for  20  years  the  face  of  the 
policy  will  be  paid  to  the  insured.  A  man  aged 
25  buys  a  $5,000.  pohcy  of  this  kind  agreeing 
to  pay  $49.25  per  $1,000.  annually,  and  he  Uved 
to  collect  the  amount  of  the  policy: 
(a)  How  much  did  he  pay  in  premiums? 
(VIII-137) 


ARITHMETIC 

(h)  How  much  did  he  collect? 

(c)   How  old  was  he  when  he  collected? 

3.  A  certain  policy  stipulates  that  after  the  premium 

has  been  paid  for  20  years  no  further  payments 
need  be  made,  but  that  at  the  time  of  the  death 
of  the  insured  the  face  of  the  policy  shall  never- 
theless be  paid  to  the  beneficiary.  A  man  aged 
35  bought  a  $10,000.  policy  on  this  plan  at 
$38.25  annually  per  $1,000.,  and  lived  to  the 
age  of  73. 
(a)  How  much  was  paid  as  premium  on  this  policy? 
(6)  How  much  was  paid  to  the  beneficiary  at  the 

time  of  this  man's  death? 
(c)   Allowing  6%  simple  interest,  how  much  less 
would  have  been  the  insurance  company's 
profit  on  this  policy  had  this  man  died  at 
the  age  of  63? 

4.  A  man  paid  an  annual  premium  of  $15.00  on  an 

accident  insurance  policy  which  provides  for  the 
payment  of  $30.00  per  week  during  disability 
caused  by  accident.  If  this  man  was  disabled 
for  10  weeks  after  having  paid  the  premium  for 
8  years,  how  much  did  this  policy  save  him? 

5.  A  manufacturer  bonded  his  cashier  for  $5,000., 

paying  therefor  an  annual  premium  of  $5.00 
per  $1,000.      After  20  years  of  faithful  work, 
this   cashier   became   addicted   to   the   use   of 
liquor  and  embezzled  $2,000. 
(a)  Without  considering  interest,  how  much  did 
this  manufacturer  gain  or  lose  by  having 
had  this  insurance? 
(VIII-138) 


PERCENTAGE 

(6)   How  much  did  the  bonding  company  gain  or 
lose? 
6.  A  steel  manufacturer  bought  an  employers'  lia- 
bility insurance  policy  at   an   annual  cost   of 
$875.  under  the  terms  of  which  the  insurance 
company's  liability  was  limited  to  $10,000.  on 
account  of  injury  to  any  one  person,  or  $30,000. 
on    account    of    injuries    caused    by   any  one 
accident.  After  paying  the  annual  premium  for  5 
years,  a   serious  accident  caused  injuries  to   a 
number  of  workmen,  who  under  the  Workmen's 
Compensation   Law    were    entitled   to    receive 
$45,860.,  of   which  amount  not   over  $10,000. 
was  due  any  one  man. 
(a)  Without  considering  interest,  how  much  did 
the  manufacturer  gain  by  having  this  insur- 
ance? 
(6)  How  much  did  the  insurance  company  lose? 

(c)  How  much  of  the  $45,860.  did  the  employer 

have  to  pay? 

(d)  How  much  did  the  insurance  company  have 

to  pay? 

Exercise  71 — Oral  Review. 

See  if  every  child  can  ask  and  answer  a  question  on 
this  today.     Somebody  else  must  ask  you  a  question. 

1.  How  do  we  find  the  area  of  a  circle?    Of  a  triangle? 

Of  a  trapezoid?     Of  a  parallelogram? 

2.  How  do  we  find  the  volume  of  a  cube?     Of  a 

prism?      Of  a  pyramid?      Of  a  cone?      Of  a 
cylinder?     Of  a  sphere? 

(VIII-139) 


ARITHMETIC 

3.  How  do  we  find  the  area  of  the  lateral  surface  of 

a  cyhnder?  Of  a  prism?  Of  a  pyramid?  Of 
a  cone?  Of  a  frustum  of  a  cone?  Of  a  frustum 
of  a  pyramid?     Of  a  sphere? 

4.  What  single  word  means  1,000  meters?     100  liter? 

10,000  grams?  10  meters?  1  Hter?  .1  gram? 
.01  meter?     .001  liter? 

5.  What  is  the  unit  of  volume  in  the  metric  system? 

Of  weight?    Of  capacity?    Of  length?    Of  area? 

6.  How  many  degrees  are  there  in  a  semi-circle?     In 

a  right  angle? 

7.  What  is  the  principal  difference  between  a  bond 

and  a  share  of  stock?  Why  are  government 
bonds  issued  at  a  lower  rate  of  interest  than 
other  bonds? 

8.  A  corporation's  net  profits  are  several  times  as 

great  as  the  dividends  on  its  preferred  stock; 
would  you  rather  own  10  shares  of  the  pre- 
ferred stock  or  of  the  common  stock  of  this 
corporation?     Wh}^? 

9.  Is  a  stockholder  a  creditor  of  a  corporation?     Is 

a  bondholder?  In  case  of  the  failure  of  a  cor- 
poration would  you  rather  be  one  of  its  bond- 
holders or  one  of  its  stockholders?  \\Tiy? 
10.  What  is  the  difference  in  time  between  two  points 
if  they  are  separated  by  90°  of  longitude?  By 
180°?     By  1°?     By  17    By  1"? 

Exercise  72 — Written  Review. 

1.  Reduce  36,872"  to  degrees,  minutes,  and  seconds. 

2.  (a)  48^  =  ?     (6)  643  =  9    

3.  (a)  V670,761  =  ?     (6)  Vl8^  =  ? 

(\ail-140) 


PERCENTAGE 

4.  The  legs  of  a  right  triangle  are  20  ft.  long;   what 

is  the  length  of  the  hypotenuse? 

5.  The  radius  of  a  circle  is  3  ft.  6  in.;   what  is  the 

length  of  45°  of  its  circumference? 

6.  The  diameter  of  a  circle  is  10  in.;  what  is  the  area 

of  a  sector  measuring  135°  of  this  circle? 

7.  What  is  the  volume  of  a  pyramid  which  has  a 

24-ft.  square  for  its  base,  and  whose  altitude  is 
20  ft.?     What  is  its  slant  height? 

8.  A  furniture  dealer  advertised  as  follows:    ''We 

furnish  four  rooms  complete  for  $99.00  payable 
$4.00  cash  and  $5.00  per  month."  If  he  charged 
6%  interest  on  every  sale  of  this  kind,  how  much 
interest  would  he  collect  on  16  sales? 

9.  The  proceeds  of  a  note  discounted  for  90  days  at 

6%  amounted  to  $665.86;   what,  was  the  princi- 
pal of  the  note? 
10.  What  is  the  yield  on  a  15-year  5%  $1,000.  bond 
bought  at  92  f,  allowing  |%  for  brokerage? 

Copy  and  divide: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 


11.  11,648 


416;  14.  18,936  --  36; 


12.  19,822  --  374;  15.  33,746  ^  47; 

13.  37,368  -^  519;  16.  59,156  ^  92. 

Subtract,  but  do  not  copy: 

(Time  for  these  12  examples  is  less  than  4  minutes.) 
17.  18.  19.  20. 

375,412  639,741  712,041  864,342 

241,987  347,953  319,751  716,875 

(VIII-141) 


ARITHMETIC 

21. 
931,204 
156,287 

22. 

728,041 
319,876 

23. 

413,841 
148,639 

24. 

286,319 
196,056 

25. 

741,093 
359,749 

26. 

612,058 
295,069 

27. 

761,432 
439,357 

28. 

870,002 
691,587 

Copy  and  multiply: 

(Time  for  these  8  examples  is  less  than  4  minutes.) 


29.  519  X  643; 

30.  809  X  592; 

31.  628  X  550; 

32.  347  X  724; 


33.  437  X  585 

34.  648  X  234 

35.  541  X  672 

36.  708  X  509. 


Add,  but  do  not  copy: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 


37. 

5,328 
7,639 
9,348 

8,985 
7,982 
2,348 


38. 

3,639 

8,723 
8,529 
3,987 
4,893 
2,789 


39. 

6,759 
3,423 

8,979 
7,598 
4,381 
1,975 


40. 

5,189 
1,939 
2,579 
9,536 
3,870 
4,797 


41. 

8,643 
9,075 
7,638 
4,928 
5,632 
6,789 


42. 

1,898 
3,990 
3,853 
6,492 
3,658 
7,842 


(VIII-142) 


PARTNERSHIP 

Lesson  33 

Division  of  Profits  and  Losses 


(Let   the   class  transact   this  example — one  half  of 

class  representing  A's  interest,  the  other  half  B's.) 

EXAMPLE:     A  and   B  are  partners  sharing  profits  or  losses 

according  to  their  several  investments;    A  has 

S12,000.  mvested,  and  B  has  S18,0(X).  invested ; 

their  profits  during  a  certain  year  are  S4,000. ; 

how  much  of  the  profit  should  each  partner 

receive? 

A's  investment =  §12,000.,  or    40% 

B's  investment =    18,000.,  or    60% 

Total  investment =  S30,000.,  or  100% 

A's  share  of  profits =    40%  of  S4,000.,  or  Sl,600. 

B's  share  of  profits =    60%  of  $4,000.,  or    2.400. 

Total  profits =  100%  S4,000. 


A  partnership  is  the  association  of  two  or  more  per- 
sons for  the  purpose  of  carrying  on  an  enterprise  or 
business  according  to  agreement. 

The  agreement  which  the  partners  enter  into,  and 
which  stipulates  what  the  relations  between  the  part- 
ners shall  be,  how  much  each  shall  invest,  how  the 
profits  or  losses  shall  be  divided,  how  long  the  agree- 
ment shall  be  in  force,  etc.,  is  called  the  '^partnership 


agreement. 


A  partnership  dissolves  upon  the  expiration  of  the 
partnership  agreement,  upon  the  death  of  one  of  the 
partners,  or  by  mutual  consent  of  the  partners. 

(Vni-143) 


ARITHMETIC 

Profits  or  losses  are  sometimes  divided  in  proportion 
to  the  amount  invested  by  each  of  the  several  partners, 
and  sometimes  according  to  a  fixed  basis,  as  50%  each 
if  there  are  two  partners,  33  J%  each  if  there  are  three 
partners,  etc. 


(Two  rows  can  represent  A,  two  rows  B 
and  two  rows  C.  Watch  j-our  money.) 
EXAMPLE:     A  owns  f  of  a  business,  and  B  o-^tis  |  of  it;   they 
decide  to  sell  |  interest  in  the  business  to  C; 
what  part  of  the  business  wiU  each  partner  own 
after  C  is  admitted  to  partnership? 
Before  the  Sale  The  Sale  After  the  Sale 

A's  share  =  |      |  of  A's  share  =  2?      A's  share  =  M  (I  ~  i'4) 
B's  share  =  f      3  of  B's  share  =  -^     B's  share  =  /^  (|  —  2?) 
C's  share  =_0       Total  C  buys  =  '^     C's  share   =  ^  ^i^  +  ii) 
Total .  .  .    =  I  Total ...    =  || 

(Note:    The  ratio  of  A's  share  to  B's  share  is 
5  to  3  both  before  and  after  making  the  sale. ) 


Exercise  73 — Oral. 

1.  What  is  the  association  of  two  or  more  persons  for 

the  purpose  of  carr^dng  on  an  enterprise  or  busi- 
ness called? 

2.  ^\Tiat  do  partners  do  on  entering  into  business  so 

that  there  may  be  no  dispute  regarding  the 
amount  of  money  each  is  to  invest,  how  the 
profits  shall  be  divided,  etc.? 

3.  \\Tiat  name  is  given  to  the  agreement  between 

partners? 

4.  Name  three  ways  in  which  a  partnership  dissolves? 

5.  What  is  meant  when  we  say  that  profits  or  losses 

are  to  be  divided  in  proportion  to  each  partner's 
investment? 

rVTII-144) 


PARTXERSHIP 

6.  "^liat  is  meant  when  we  say  that  profits  or  losses 

are  to  be  divided  according  to  a  fiixed  basis? 

7.  If  it  is  agreed  between  tliree  partners  that  A  is 

to  receive  ^  of  the  profits.  B  f  and  C  |,  is  this 
a  case  of  di^dding  the  profits  according  to  a  fixed 
basis  or  in  proportion  to  the  investments  of  the 
partners? 

8.  If  A  has   86,000.   invested  and  B  has   84,000. 

invested,  and  they  divide  the  profits  on  the  basis 
of  60^  and  40^-.  is  this  a  case  of  di\'iding  the 
profits  according  to  a  fixed  basis  or  in  proportion 
to  the  iavestments  of  the  paitners? 

9.  Henry  and  Frank  are  partners  deahng  ia  apples: 

Hemy  has  3  apples  and  Frank  has  6  apples: 
they  decide  to  sell  f  of  their  apples  to  William 
for  9c;  how  much  of  this  9c  should  each  boy 
receive? 
10.  In  Question  9.  how  many  apples  ^iU  each  boy 
have  after  the  sale  is  made? 

Exercise  74 — Written. 

1.  Frederick  Jolmson  and  George  Brown  are  partners 

shariQg  profits  and  losses  on  the  basis  of  Johnson 
60^-  and  BroT^Ti  40^:  for  a  certain  year  their 
gross  earnings  were  849,365.  and  their  expenses 
were  840,520. :  what  was  each  partner's  share  of 
the  net  profit? 

2.  Robt.  -Alexander,  Benj.  Wilson,  and  Chas.  Hend- 

ricks are  partners  sharing  profits  and  losses  on 
the  basis  of  40^,  35^,  and  25*^  respectively: 
for   a    certain   year  their   gross   earmngs   were 


ARITHMETIC 

$31,612.35  and  their  expenses  were  $34,729.35; 
what  was  each  partner's  share  of  the  loss? 

3.  Henry   Black   and   George   White   are   partners 

sharing  profits  and  losses  in  proportion  to  their 
investment;  at  the  end  of  a  certain  year  they 
find  that  they  have  made  a  profit  of  $8,634.60; 
what  is  each  partner's  share  of  this  profit  if 
Black's  capital  account  showed  he  had  $11,748.00 
invested  and  White's  capital  account  showed  he 
had  $9,612.00  invested?     (Be  very  accurate.) 

4.  From  the  following  information  find  each  partner's 

share  of  the  net  profit  or  loss  for  the  year,  the 
arrangement  between  the  partners  being  that  h 
of  the  profits  are  to  be  divided  equally  between 
them,  the  other  J  being  divided  in  proportion 
to  their  investments : 

Gross  Earnings. .  .$36,400. 

Expenses 28,200. 

A's  Capital 18,000. 

B's  Capital 12,000. 

5.  A  and  B  are  partners  in  business;    their  capital 

accounts  show  that  A  has  $6,552.  invested  and 
B  has  $6,048.  invested;  they  decide  to  admit  C 
as  a  partner  with  a  25%  interest;  what  share  of 
the  business  will  each  partner  own  after  C  is 
admitted? 

6.  Frank  Powell  has  $32,000.  invested  in  business 

when  he  decides  to  admit  Walter  Pearson  as  a 
partner;  Pearson  is  to  invest  an  amount  which 
shall  be  60%  of  their  combined  capital;  how 
much  did  Pearson  invest? 

(VIII-146) 


PARTNERSHIP 

7.  X  and  Y  are  partners  in  business;  X's  investment 
is  $4,480.;  Y's  investment  is  $3,680.;  they 
decide  to  bring  their  capital  accounts  to  a  50% 
—  50%  basis  by  investing  or  withdrawing  as  the 
case  may  be;  what  amount  does  each  partner 
invest  or  withdraw? 

Exercise  75 — Written. 

A  Grocery  Partnership 

1.  A  and  B  are  partners  in  the  grocery  business; 

A's  investment  is  $4,824.;  B's  investment  is 
$3,216.;  they  decide  to  equahze  their  capital 
accounts  by  investing  or  withdrawing  as  the  case 
may  be;  what  amount  must  A  invest  or  with- 
draw? 

2.  T\Tiat  amount  must  B  invest  or  withdraw? 

3.  They  decide  to  admit  C  as  a  partner  with  J  inter- 

est, he  to  invest  a  sum  similar  to  that  of  each 
of  the  two  other  partners;  what  amount  must 
C  invest? 

4.  At  the  end  of  the  year  they  found  that  their  gross 

earnings  Vv^ere  $36,000.  and  their  expenses  were 
$24,000. ;   what  was  each  partner's  net  profit? 

5.  At  the  end  of  the  second  year  their  capital  accounts 

stood  as  follows: 

A    $4,040.;        B     $3,020.;        C     $2,940.; 

They  decided  to  discontinue  business  at  this 
time  and  after  selling  all  of  their  merchandise 
and  other  property  and  settling  all  of  their  debts, 
they  had  exactly  $8,000.  cash  left;  how  much 
did  they  lose  in  selling  out? 

(VIII-147) 


ARITHMETIC 

6.  How  much  of  this  loss  should  be  charged  to  each 

partner? 

7.  How  much  of  the  $8,000.  cash  belongs  to  A? 

8.  How  much  belongs  to  B? 

9.  How  much  belongs  to  C? 

10.  After  paying  the  proper  amount  to  each  of  the 
partners,  what  balances  would  appear  on  the 
partners'  accounts? 

Exercise  76 — Oral  Review. 

1.  Read:  (a)  48,874,936,724;  (6)  473,000,001,002. 

2.  What  is  the  cost  of  72  dozen  eggs  at  33 1^  per 

dozen? 

3.  If  87  J  lb.  of  butter  cost  $28.00,  what  is  the  cost 

per  lb? 

4.  Add,  and  read  the  answers  at  sight: 

(a)  (h)  (c) 

4,634  3,872  9,726 

7,362  1,498  1,265 

5.  Subtract,  and  read  the  answers  from  left  to  right: 

(a)  (6)  (c) 

974  882  6,321 

699  595  895 

6.  What  is  the  interest  on  $60.00  at  6%  for  293  days? 

7.  How  much  wages  will  a  boy  earn  if  he  works  37 

hours  at  $12.00  per  week  on  a  48-hour  basis? 

8.  Draw  the  following  or  hold  the  object  in  your 

hand  before  the  class  and  make  an  interesting 
talk  on  each: 

(VIII-148) 


PARTNERSHIP 


I 


EXAMPLE:  A  Square. 
This  is  a  square  because  it  is  an  area  bounded  by  four  straight 
equal  sides  and  all  its  angles  are  right  angles.  One  side  may 
be  the  length  or  width.  The  sum  of  the  four  sides  is  the  peri- 
meter. Area  is  found  by  multiplying  the  number  of  square  units 
in  1  row  by  the  number  of  rows,  or  area  =  length  X  width. 
If  area  is  given  extract  the  square  root  to  get  one  side.  To 
find  diagonal,  use  Pythagorian  Theorem ;  H^  =  A^  +  B^;  H  = 


A2  +  B2. 

(a)  A  square; 

(h) 

A  cube; 

(h)  An  oblong; 

ii) 

A  prism; 

(c)   A  rectangle; 

(i) 

A  cylinder; 

(d)  A  triangle; 

(A:) 

A  cone; 

(e)   A  trapezoid; 

(0 

A  pyramid ; 

(/)  A  parallelogram; 

(m) 

A  sphere; 

(g)   A  circle; 

in) 

A  frustum. 

9.  Be  ready  to  define  the  following: 


(a)   Simple  Interest; 
(6)  Bank  Discount; 

(c)  Trade  Discount; 

(d)  Cash  Discount; 

(e)  Premium  and  Discount 


(/)  Insurance  Premium; 

(g)  Commission; 

(h)  Brokerage; 

(i)    Taxes ; 

(j)  Compound  Interest. 


10.  Be  ready  to  give  a  clear  idea  of   the   following 
topics : 

(a)  A  bond ; 

(6)  A  share  of  stock; 

(c)  A  corporation; 

(d)  A  stockholder; 

(e)  A  bondholder; 


(/)  Partnership; 


(9) 
(h) 

U) 
(k) 

(I) 
(VIII-149) 


Dividends ; 

Par  value  of  stock; 

Market  value  of  stock; 

Yield; 

Assets ; 
Liabilities ; 


ARITHMETIC 

(m)  Capital;  (o)    Installments; 

(n)   Lien;  (p)   Mortgage. 

Exercise  77 — Written  Review. 

1.  What  is  the  interest  on  $438.00  from  Aug.  8,  1921, 

to  May  8,  1922,  at  5%? 

2.  What  is  the  area  of  the  largest  circle  which  can  be 

drawn  in  a  square  whose  area  is  400  sq.  in.? 

3.  What  is  the  hypotenuse  of  a  right  triangle  if  each 

of  its  legs  is  9"  in  length? 

4.  What  is  the  altitude  of  an  isosceles  triangle  whose 

base  is  6"  and  whose  equal  sides  are  5"  long? 

5.  The  difference  in  longitude  between  Chicago  and 

New  York  is  approximately  15°;  when  it  is  4 
p.  M.  in  Chicago,  what  time  is  it  in  New  York? 

6.  A  $500.  bond  bears  6%  interest  payable  Jan.  1st 

and  July  1st  of  each  year;  how  much  cash  would 
be  required  to  buy  this  bond  at  97  and  accrued 
interest  on  April  1st?     (Omit  brokerage.) 

7.  Find  the  diagonal  of  a  6-inch  cube. 

8.  A  phonograph  is  sold  for  $200.00  on  terms  of 

$20.00  cash  and  $10.00  per  month  plus  6% 
interest;  what  is  the  interest  on  this  installment 
account? 

9.  V4,637.61  =  ^ 

10.  What  is  the  length  of  one  side  of  a  square  which 

has  an  area  of  1,604.8036  sq.  yd.? 

11.  Express  the  values  of  the  following  sales: 

100  -  $1,000.  Liberty  Bonds  -  *? 
2,000  -  $500.  Liberty  Bonds  =  ? 
2,000  -     $100.  Liberty  Bonds  =  ? 

(VIII-150) 


PARTNERSHIP 


1,000  -      SoOO.  Liberty  Bonds  =  ? 

60,000  -       $50.  Liberty  Bonds  -  ? 

800,000  -     $100.  Liberty  Bonds  =  "? 

50,000  -  $5,000.  Liberty  Bonds  =  "? 

20,000,000  -     $100.  Liberty  Bonds _=_^ 

Total  value =  ? 

12.  (a)  This  value  is  approximately  what  %  of  the 
total  issue  of  $6,000,000,000.? 
(6)  The  last  sale  is  what  %  of  the  total  issue? 
(c)    If  the  total  amount  subscribed  was  $8,246,- 
523,000.  the  issue  was  oversubscribed  by  what 
%  approximately? 

Copy  and  multiply: 

(Time  for  these  8  examples  is  less  than  4  minutes.) 


13.  681  X  72 

14.  558  X  27 

15.  782  X  22 

16.  888  X  24 


17.  706  X  62; 

18.  784  X  36; 

19.  973  X  84; 

20.  738  X  74. 


Subtract,  but  do  not  copy: 

(Time  for  these  12  examples  is  less  than  4  minutes.) 
21.  22.  23.  24. 

539.285  621,741  397,412  867,412 

197.286  387,219  297,468  382,941 


25. 
100,874 
99,946 

26. 

631,025 
41,973 

27. 

612,318 
200,493 

28. 

298,741 

289,765 

29. 
300,412 
299,975 

30. 

538,763 
494,839 

31. 

658,729 
597,831 

32. 

284,302 
178,463 

(VIII- 

-151) 

ARITHMETIC 

Copy  and  divide : 

(Time  for  these  6  examples  is  less  than  4  minutes.) 

33.  11,286  ^  38;  36.  83,996  ~  92; 

34.  30,076  -^  73;  37.  60,900  -^  84; 

35.  24,769  ~  47;  38.   16,120  -^  65. 

Add,  but  do  not  copy: 

(Time  for  these  6  examples  is  less  than  4  minutes.) 


39. 

40. 

41. 

42. 

43. 

44. 

12,874 

8,649 

38,741 

5,196 

6,641 

7,639 

319 

42,487 

349,965 

863 

45,875 

148,759 

6,487 

29 

416 

5,887 

93,241 

213 

29 

538 

3,148 

43,971 

1,859 

6,874 

413,528 

7,625 

40,329 

6,329 

7,612 

58 

4,196 

5,387 

7,588 

9,658 

9 

45,812 

(Vni-152) 


DEFINITIONS 
(Parts  I  to  VIII,  Inclusive.) 

Abstract  Number A  number  which  is  used  without  the  name  of 

an  object. 
Account A  record  sho\^^ng  all  business  transactions  with 

any  person. 
Acute  Angle An  angle  which  is  sharper  (or  less)  than  a  right 

angle. 
Addends The  numbers  which  are  to  be  added  in  an 

example  in  addition. 
Addition Uniting  two  or  more  numbers  or  quantities 

into  one  number  or  quantity. 
The  numbers  to  be  added  are  called  "addends." 
The  answer  is  called  the  "sum"  or  "total." 
The  sign  of  addition  is  +,  which  is  called  "  plus." 
Agent (In  Commission.)     The  person  who  is  engaged 

by  the  principal  to  perform  some  service. 

AUquot  Part An  equal  part  of  a  number. 

Altitude Height. 

Amount (In  Interest.)     The  principal  plus  the  interest. 

Angle The  opening  between  two  straight  lines  which 

meet  in  a  point. 

Angular  Measure The  table  of  measures  used  for  measuringangles. 

Apex The  highest  point  or  summit  of  a  plane  or  soUd 

figure. 
Apothecaries'  Dry 

Measure The  table  of  measures  used  by  druggists  and 

physicians  in  measuring  dry  chemicals. 
Apothecaries'  Liquid 

Measure The  table  of  measures  used  by  druggists  and 

physicians  in  measuring  liquid  chemicals. 
Apothem A  perpendicular  line  drawn  from  the  center  of 

one  of  the  sides  to  the  center  of  a  many- 
sided  figure. 

Arabic  Numerals The  numbers  in  common  use,  as  1,  2,  3,  etc. 

Arc Any  part  of  a  circumference  less  than  the  whole, 

CVTII-1.53) 


AHITrmETIC 

Area 4 The  space  within  the  HmitiS  or  boundaries  of  a 

surface. 

Arithmetic The  science  of  numbers. 

A^'erage A  medium  number  which  can  be  used  in  place 

of  each  of  several  unequal  numbers. 
Avoirdupois  Weight.  .  .  .The  table  of  weights  used  for  weighing  all  com- 
mon articles,  such  as  groceries,  meats,  hay,  etc. 
Balance The  difference  l^etween  the  two  sides  of  an 

account. 
Bank  Discount Interest  deducted  in  advance  by  a  bank  from 

the  amount  of  a  note. 
Ba.?e (In  Mensuration.)      The  side  of  a  figure  on 

which  the  figure  appears  to  rest. 
Base (In  Percentage.)     The  number  or  quantity  on 

which  the  percentage  is  computed. 

Bill See  '' Sales-slip." 

Billions'  Period The  period  of  the  4th  rank,   counting  units' 

period  as  the  first. 
Board  Measure The  unit  of  measure  used  for  measuring  boards 

isthe"  Board  Foot,  "which  equals  1'  XI'  Xl". 
Bond An  evidence  of  indebtedness  usually  secured  by 

a  mortgage. 
Brackets Marks  used  to  enclose  numbers  which  are  to 

be  treated  as  one  number.     (  ) ;    [  ] ;    |    | . 
Broker One  who  sells  stocks,  bonds,  real  estate,  etc., 

on  a  percentage  basis. 

Brokerage The  compensation  paid  to  a  broker. 

Buyer One  who  bu3's. 

Cancellation Reducing  the  numerator  of  one  fraction  and  the 

denominator  of  another  to  simplify  multi- 
plication. 
Capital  Stock The  total  amount  for  which  a  corporation  can 

issue  stock  certificates  as  specified  in  its  charter. 
Carry To  convert  10  of  any  order  into  1  of  the  next 

higher  order. 

Cash  Account An  account  with  Cash. 

Cash  Book The  book  in  which  the  Cash  Account  is  kept. 

Cash  Discount An  amount  allowed  for  the  payment  of  a  bill 

on  or  before  a  certain  date. 
Change. To  convert  1  of  any  order  into  10  of  the  next 

lower  order. 

(Vni-1.54) 


I 


DEFINITIONS 

Check An  order  drawn  on  a  bank  to  pay  a  sum  of 

money  from  funds  on  deposit. 
Circle A   plane  figure    bounded   by   one   continuous 

curved  line  which  at  all  points  is  a  uniform 

distance  from  a  point  in   the  center  of  the 

figure. 
Circular  Measure The  table  of  measures  used  for  measuring  circles 

and  arcs. 
Circumference The  distance  around  a  circle.     The  line  which 

forms  the  boundary  of  a  circle. 
Commercial  Interest .  .  .  Interest  computed  on  the  basis  of  360  days  to 

the  year. 
Commission An  amount  of  money  paid  by  one  person  who 

is  called  the  ''principal"  to  another  person 

who  is  called  the  ''agent,"  for  some  service 

the  agent  performs  for  the  principal. 
Common  Fraction A  fraction  of  which  both  the  numerator  and  the 

denominator  are  written. 
Common  Stock Stock  on  which  unlimited  dividends  may  be 

paid   after   preferred   dividends   have   been 

provided  for. 

Common  Year A  year  containing  365  days. 

Complement The  difference  between  a  number  and  10,  100, 

1,000,  etc. 
Complex  Fraction A  fraction  containing  a  fractional  denominator 

or  numerator. 
Composite  Number ....  Any  number  which  is  composed  of  two  or  more 

factors. 
Compound  Denominate 

Number A  denominate  number  which  consists  of  two  or 

more  denominations. 
Compound  Fraction. ...  A  fraction  of  a  fraction. 
Compound  Interest ....  When  interest  for  stated  periods  is  added  to  the 

principal  and  the  amount  so  found  is  used  as 

the  principal  for  the  next  interest  period,  the 

total  interest  is  called  "Compound  Interest." 
Compound  Proportion  .  .  A  proportion  consisting  of  more  than  two  pairs 

of  terms. 
Concrete  Number A  number  which  is  used  with  the  name  of  an 

object. 

(Vni-155) 


ARITHMETIC 

Cone A  solid  having  a  circular  base  and  one  curved 

side  which  tapers  uniformly  from  the  base  to 
a  vertex  directly  over  the  center  of  the  base. 

Consecutive  Numbers .  .  Numbers  which  follow  one  another  without 

interruption  of  any  kind;  as  1,  2,  3,  4,  5,  etc. 

Consignment A  shipment  of  merchandise  to  be  sold  for  the 

account  of  the  shipper. 

Corporation An  artificial  body  created  and  chartered  by  the 

law  of  the  state  to  transact  its  specific  busi- 
ness as  if  it  were  an  individual. 

Counting To  name  one  by  one  to  find  the  number  of  units 

in  a  group. 

Credit. An  entry  which  shows  that  the  person  or  thing 

on  whose  account  the  entry  appears,  has 
parted  with  sometliing  of  value. 

Creditor One  to  whom  money  is  owing. 

Cube (In  Mensuration.)     A  solid  having  six  square 

sides  or  faces  of  equal  size,  joined  so  that 
every  angle  is  a  right  angle. 

Cube (In  Powers  and  Roots.)      The  power  of  the 

third  degree  of  any  number. 

Cube  Root The  root  of  a  power  of  the  third  degree. 

Cubic  Contents The  space  occuped  by  a  sohd.     (Volume.) 

Cubic  Measure The  table  of  measures  used  for  measuring  the 

cubic  contents  or  volume  of  soHds. 

Cylinder A  solid  bounded  by  a  uniformly  curved  side 

and  two  parallel  circular  ends  or  bases  of 
equal  size. 

Date  of  Maturity The  date  on  which  a  note  or  other  obhgation 

matures  or  becomes  due. 

Debit An  entry  which  shows  that  the  person  or  thing 

on  whose  account  the  entry  appears,  has 
received  something  of  value. 

Debtor One  who  owes  money. 

Decillions'  Period The  period  of  the  12th  rank,  counting  units' 

period  as  the  first. 

Decimal Numbered  by  tens. 

Decimal  Fraction (More  commonly  called  "Decimal.")  A  frac- 
tion with  a  denominator  of  10,  100,  etc.,  the 
denominator  being  indicated  by  writing  the 
numerator  to  the  right  of  a  decimal  point. 

(VIII-156) 


DEFINITIONS 

Decimal  Point A  point  or  period  (.)  used  to  indicate  that  the 

numbers  written  to  the  right  of  it  form  a 
decimal  fraction. 

Denominate  Number.  . .  A  number  used  with  the  name  of  a  measure. 

(See  also:  "Simple  Denominate  Number" 
and  "Compound  Denominate  Number.") 

Denominator That  term  of  a  fraction  which  shows  into  how 

many  equal  parts  a  unit  has  been  divided. 

Diameter The  distance  across  a  circle  through  the  center. 

Difference The  answer  found  by  subtraction. 

Digit Any  single  figure,  as  1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

Dimension The  extent  of  a  line,  area,  or  solid. 

Disbursements Amounts  of  money  paid  out. 

Discount A  reduction  from  the  regular  price  of  an  article 

for  any  reason.  (See  also:  "Cash  Discount," 
"Trade  Discount,"  and  "Bank  Discount."). 

Discount The  difference  between  the  par  value  and  the 

market  value  of  stocks  and  bonds  when  the 
par  value  is  the  greater. 

Divided  by  (-h) The  sign  of  division. 

Dividend The  number  to  be  divided  in  an  example  in 

division. 

Dividend A  distribution  of  profits  by  a  corporation. 

Division Finding  how  many  times  one  number  is  con- 
tained in  another  number,  and  finding  one 
of  the  equal  parts  of  a  number. 
The  number  to  be  divided  is  called  the  "divi- 
dend." 
The  number  which  shows  into  how  many  parts 
the  dividend  is  to  be  divided,  or  which  shows 
the  size  of  the  parts  when  the  number  of 
parts  is  to  be  found,  is  called  the  "divisor." 
The  divisor  is  the  number  you  divide  by. 
The  answer  is  called  the  "quotient." 
The  sign  of  division  is  -^,  called  "divided  by." 

Divisor The  number  which  shows  into  how  many  parts 

the  dividend  is  to  be  divided,  or  which  shows 
the  size  of  the  parts  when  the  number  of 
parts  is  to  be  found,  in  an  example  in  division. 
The  number  you  divide  by. 

(VIII-157) 


ARITHMETIC 

Dry  Measure The  table  of  measures  used  in  measuring  grains, 

fruit,  vegetables,  etc.      (See  also:    "Apothe- 
caries' Dry  Measure.") 

Due  Date See  "Date  of  Maturity." 

Elapsed  Time The  time  between  two  dates. 

Em The  square  of  a  type  face. 

Endorsement Any  writing  on  the  back  of  a  note,  check,  etc. 

Equals  (  =  ) The  sign  of  equality. 

Equation A    statement    showing    the    equality    of    two 

quantities,  one  of  which  is  placed  before  and 

one  after  an  equal  sign. 
Equator A  line  running  completely  around  the  earth 

midway  between  the  North  Pole  and  the 

South  Pole.     The  line  from  which  latitude 

is  computed. 
Equilateral  Triangle. . .  .  A  triangle  having  three  equal  sides. 

Equivalent Having  the  same  value. 

Even  Numbers All  numbers  divisible  by  2  without  remainder; 

therefore,  all  numbers  with  0,  2,  4,  6,  and  8, 

in  units'  place. 
Exact  Interest Interest  computed  on  the  basis  of  365  days  (in 

Leap  Year  366  days)  to  the  year. 

Exponent The  number  which  indicates  the  degree  of  power. 

Extremes The  first  and  fourth  terms  of  a  proportion. 

Factor Any   number  which  is   contained  in  another 

number  without  a  remainder. 
Fraction A  number  which  shows  one  or  more  of  the  equal 

parts  of  a  unit.       (See   also:    "Compound 

Fraction"  and  "Complex  Fraction.") 
Frustum The  lower  part  of  a  pyramid  or  of  a  cone  which 

remains  after  a  plane  has  been  cut  through 

parallel  to  the  base. 

Grand  Total The  sum  of  several  totals. 

Graph  or  Graphic  Chart .  Any  drawing  or  chart  which  renders  possible 

the  visualization  of  comparative  statistics  or 

other  information. 
Great  Circle  of  a  Sphere. .  A  circle  made  by  cutting  a  sphere  into  two  equal 

parts  b}'  a  plane  passing  through  the  center. 
Greatest  Common 

Divisor The  largest  number  which  will  divide  two  or 

more  numbers  without  remainder. 

(Vni-158) 


DEFINITIONS 

(The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  also  the  Highest  Common  Factor 
of  those  numbers.) 

Gross  Price The  price  of   an  article  before  deducting  any 

discount.     The  "List  Price." 

Gross  Profit The  amount  of  profit  reahzed  from  the  sale  of 

an  article  without  making  any  deduction  for 
such  expenses  as  rent,  light,  etc. 

Group Several  persons  or  things. 

Highest  Common 

Factor The  largest  factor  which  is  common  to  two  or 

more  numbers.  (The  Highest  Common 
Factor  of  any  two  or  more  numbers  is  also  the 
Greatest  Common  Divisor  of  those  numbers.) 

Hexagon A  six-sided  polygon. 

Horizontal Parallel  to  the  surface  of  the  earth. 

Hundreds  of  ^Millions' 

Place The  place  of  the  9th  order. 

Hundreds  of 

Thousands'  Place   ,  .  .  The  place  of  the  6th  order. 

Hundreds'  Place The  place  of  the  3d  order. 

Hundredths'  Place The  2d  place  to  the  right  of  the  Decimal  Point. 

Hundred-Thousandths' 

Place The  fifth  place  to  the  right  of  the  Decimal  Point. 

Hypotenuse The  side  of  a  right  triangle  which  joins  the 

two  legs. 

Improper  Fraction A  fraction  which  means  1  unit  or  more  than  1  unit. 

Index The  number  which  indicates  the  degree  of  root. 

Insurance A  written  contract  whereby  one  party  agrees 

to  pay  a  certain  sum  to  another  party  in  case 
of  loss  or  injury  of  a  certain  kind. 

Integer A  whole  number. 

Interest ]\Ioney  paid  or  to  be  paid  for  the  use  of  money. 

(See  also:  "Compound  Interest.") 

International  Date  Line..  A  line  passing,  with  few  exceptions,  along  the 

180th  meridian,  which  is  the  dividing  line 
between  dates. 

Inventory Merchandise  on  hand. 

Invoice , A  bill  showing  the  quantities,  price,  etc.,  of 

articles  bought  from  a  wholesaler  or  manu- 
facturer. 

(VIII-159) 


ARITHMETIC 

Isosceles  Triangle A  triangle  having  two  equal  sides. 

Land  Measure The  table  of  measures  used  for  measuring  land. 

Lateral  Area The  area  of  the  lateral  faces  or  sides  of  a  solid. 

Lateral  Faces The  sides  of  a  soHd. 

Latitude Distance  north  or  south  of  the  equator. 

Leap  Year A  year  containing   1  extra  day,  making  366 

daj^s  in  all. 
Least  Common 

Denominator. The  smallest  denominator  common  to  two  or 

more  fractions. 
Least  Common 

Multiple The  smallest  number  which  contains  two  or 

more  other  numbers  without  remainder. 

Ledger The  book  in  which  accounts  are  kept. 

Legs  of  a  Triangle The  two  sides  of  a  right  triangle  which  form 

the  right  angle. 
Linear  Measure The  table  of  measures  used  for  measuring  lines 

and    distances.       (See    also:      "Surveyors' 

Linear  Measure.") 
Liquid  Measure The   table    of    measures   used   for   measuring 

common  liquids.     (See  also:   "Apothecaries' 

Liquid  Measure.") 
List  Price The  price  at  which  an  article  is  listed  in  a 

catalogue  or  price-Hst. 
Location One  object's  or  number's  position  in  relation 

to  the  other  objects  or  numbers  in  a  group. 
Long  Division The  method  of  division  in  which  partial  divi- 
dends are  WTitten. 

Longitude Distance  east  or  west  of  the  prime  meridian. 

Loss The  difference  between  the  cost  and  the  selling 

price,  when  the  cost  is  the  greater. 
Lowest  Terms The  simplest  form  in  which  a  number  or  frac- 
tion can  be  WTitten. 
Maker The  person  who  "makes"  or  signs  a  promissory 

note. 

Manufacturer One  who  makes  or  manufactures  merchandise. 

Market  Value The  value  at  which  stocks  or  bonds  can  be  sold 

at  a  particular  time. 

Means The  second  and  third  terms  of  a  proportion. 

Mensuration Taking  the  measurements  of  anything. 

Merchandise Any  article  which  can  be  bought  and  sold. 

(VIII-160) 


DEFINITIONS 

Merchandise  Counting 

Table The  table  used  for  counting  merchandise  of 

all  kinds. 

Metric  System A   decimal  sj^stem   of   weights   and  measures 

devised  by  the  French. 

Millions'  Period MiUions',  Tens  of  Milhons',  and  Hundreds  of 

Milhons'  Places. 

MiUions'  Place The  place  of  the  7th  order. 

Millionths'  Place The  sixth  place  to  the  right  of  the  Decimal  Point. 

Minuend The  number  from  which  we  subtract  in  ai^ 

example  in  subtraction. 

Minus  ( — ) The  sign  of  subtraction. 

Mixed  Decimal A  number  which  shows  both  an  integer  and  a 

decimal  fraction. 

Mixed  Number A  number  which  shows  one  or  more  units,  plus 

one  or  more  parts  of  a  unit ;  therefore,  a  com- 
bination of  a  whole  number  and  a  fraction. 

Mortgage A  document  pledging  property  as  security  for 

the  payment  of  a  debt.     (See  also:    ''Satis- 
faction of  Mortgage.") 

Mortgagee The  person  in  whose  favor  a  mortgage  is  given, 

Mortgager The  person  who  gives  a  mortgage. 

Most  Convenient 

Denominator See  "Least  Common  Denominator." 

Multiple A  number  which  contains  another  number  more 

than  once  without  remainder. 

MultipHcand The  number  which  is  to  be  repeated  in  an 

example  in  multiplication. 

MultipHcation Finding  a  number  or  quantity  by  repeating 

another  number  or  quantity  a  given  number 
of  times. 
The  number  which  is  to  be  repeated  is  called 

the  "multiphcand." 
The  number  which  shows  how  many  times  the 
multiplicand  is  to  be  repeated  is  called  the 
"multiplier." 
The  answer  is  called  the  "product." 
The  sign  of  multiplication  is  X,  called  "multi- 
plied by"   when  the  multiplier  follows   it, 
and    "times"    when    the    multipHer    comes, 
before  it. 

(Vin-161) 


ARITHMETIC 

Multiplied  by  (X) The  sign  of  multiplication. 

Multiplier The  number  which  shows  how  many  times  the 

multiphcand  is  to  be  repeated  in  an  example 

in  multiplication. 
Net  Price The   price   of   an   article   after   deducting   all 

discounts. 
Net  Proceeds The  amount  of  money  reaHzed  from  a  consign- 
ment after  all  commissions  and  other  expenses 

are  deducted. 
Net  Profit The  final  profit  remaining  after  deducting  all 

expenses  of  every  nature. 
NoniUions'  Period The  period  of  the  11th  rank,  counting  units' 

period  as  the  first. 

Note See:  "Promissory  Note." 

Numerator > That  term  of  a  fraction  which  shows  how  man\' 

parts  of  the  unit  are  being  spoken  of. 

Oblique A  position  wliich  slants. 

Oblique  Angle An  angle  w^hich  is  either  acute  or  obtuse.     (Any 

angle  which  is  not  a  right  angle.) 
Oblong The  plane  figure  formed  by  joining  the  ends  of 

two    straight    Hnes    of    one    length    to    two 

straight  Unes  of  another  length,  so  that  they 

form  four  right  angles. 
Obtuse  Angle An  angle  which  is  blunter  (or  greater)  than  a 

right  angle. 

Octagon An  eight-sided  polygon. 

OctilUons'  Period The  period  of  the  10th  rank,  counting  units' 

period  as  the  first. 
Odd  Number All  numbers  not  divisible  by  2  without  remain- 
der;   therefore,  all  numbers  with  1,  3,  5,  7, 

or  9,  in  units'  place. 

Order The  place  occupied  by  a  number,  as : 

1st  order  is  Units'  Place. 

2d  order  is  Tens'  Place. 

9th  order  is  Hundreds  of  MilHons'  Place. 

Paper  Measure The  table  of  measures  used  in  measuring  paper. 

Parallel Running  in  the  same  direction  with  an  equal 

distance  between. 
Parallelogram Any  plane  figure  bounded  by  four  straight  lines, 

having  two  sets  of  parallel  sides. 

(VITI-1G2) 


DEFINITIONS 

Partial  Dividends The  several  dividends  necessary  in  finding  the 

quotient  in  an  example  in  long  division. 

Partial  Products The  several  products  which,  when  added,  form 

the  final  product  in  an  example  in  multi- 
plication. 

Partnership The  association  of  two  or  more  persons  for  the 

purpose    of    carrying    on    an    enterprise    or 
business  according  to  agreement. 

Par  Value The  value  of  stocks  and  bonds  as  indicated 

thereon. 

Payee The  person  in  whose  favor  a  promissory  note, 

bill  of  exchange,  or  check  is  made. 

Pentagon A  five-sided  pol3'gon. 

Per  Annimi By  the  year. 

Per  Cent By  the  hundred.     Also :  Hundredths. 

Percentage Calculation     by     hundredths.        Also:      The 

answer  of  an  example  in  percentage. 

Perimeter The  sum  of  the  sides  which  form  the  boundary 

of  a  figure. 

Period Units',  Tens',  and  Hundreds'  Places  when  con- 
sidered as  a  group. 

Pi  (Pronounced  Pi) .  .  .  .A  Greek  letter  representing  the  ratio  of  ihe 

circumference   of   a   circle  to   the  diameter 
(3.1416). 

Pica I  inch.    (Printers'  Type  Measure.) 

Place The  order  in  which  a  number  is  written,  as: 

Units'  Place  is  the  1st  order. 

Tens'  Place  is  the  2d  order. 

Hundreds  of  Millions'  Place  is  the  9th  order. 

Plus  (+) The  sign  of  Addition. 

Point 7^  inch.     (Printers'  T^^pe  Measure.) 

Polygon A  plane  figure  having  more  than  four  sides 

and  angles. 

Power The  product  obtained  by  using  any  number 

several  times  as  a  factor. 

Preferred  Stock Stock  on  which  dividends  at  a  certain  fixed  rate 

must  be  paid  before  any  other  dividends  may 
be  paid. 

Premium The  difference  between  the  par  value  and  the 

market  value  of  stocks  and  bonds,  when  the 
market  value  is  the  greater. 

CVTII-163) 


ARITHMETIC 

Premium The  amount  paid  for  insurance. 

Prime^Factor A  factor  which  cannot  itseK  be  separated  into 

other  factors. 

Prime  Meridian The  meridian  which  runs  from  pole  to  pole 

through  Greenwich  (near  London),  England, 
from  which  longitude  is  computed. 

Prime  Number Any  number  which  is  divisible  only  by  1  and 

by  itself  without  remainder. 

Principal (In  Commission.)     The  person  v/ho  engages  an 

agent  to  perform  some  service. 

Principal (In  Interest.)      The  sum  of  money  on  which 

interest  is  paid.     The  Base. 

Printers'  Type  Measure .  The  table  of  measures  used  in  measuring  type. 

Prism A    solid    having    rectangular    sides    and    two 

parallel  ends  or  bases. 

Proceeds ^ The  amount  of  a  note  remaining  after  deducting 

Bank  Discount.    (See  also :  "  Net  Proceeds.") 

Product The  answer  found  by  multiplication. 

Profit The  difference  between  the  cost  and  the  selKng 

price,  when  the  selhng  price  is  the  greater. 

Progressive  Numbers.  . .  Numbers  which  follow  one  another  in  any  regu- 
lar order,  as  2,  4,  6,  8,  etc. ;  5,  10,  15,  20,  etc. 

Promissory  Note A  promise  (in  writing)  to  pay  a  certain  sum  of 

money  at  a  certain  time. 

Proper  Fraction A  fraction  which  means  less  than  1  unit. 

Proportion The  comparison  of  equal  ratios.      (See  also : 

"Simple  Proportion"  and  "Compound  Pro- 
portion.") 

Protractor An  instrument  used  for  measuring  angles. 

Pyramid A  sohd  whose  base  is  a  triangle,   square,  or 

polygon  and  whose  sides  or  lateral  faces 
(corresponding  in  number  to  the  number  of 
sides  in  the  base)  are  triangles  meeting  at  a 
common  point  called  the  vertex. 

Quadrant One  fourth  of  a  circle,  the  figure  being  bounded 

by  an  arc  and  two  radii  which  form  a  right 
angle. 

Quadrillions' Period ....  The  period  of  the  6th  rank,   counting  units' 

period  as  the  first. 

Quintillions'  Period.  .  .  .The  period  of  the  7th  rank,   counting  units' 

period  as  the  first. 

(VIII-164) 


DEFINITIONS 

Quotient The  answer  found  by  division. 

Radius  (plural 

"Radii") A  straight  line  drawn  from  the  center  of  a 

circle  to  a  point  in  the  circumference. 

Rank See  "Location." 

Rate A  certain  per  cent  of  the  base. 

Ratio The  relation  which  one  number  or  quantity 

bears  to  another  number  or  quantity  of  the 

same  kind. 

Receipt A  paper  showing  payment  or  deUvery. 

Receipts Amounts  of  money  received. 

Rectangle Anj^  plane  figure  which  is  bounded  by  four 

straight  sides,  and  has  four  right  angles. 

Rectangular  Prism See  "Right  Prism." 

Rectangular  Sohds Cubes  and  Right  Prisms. 

Reduction Changing  the  form  of  a  number  or  fraction 

without  changing  its  value. 

Remainder The  answer  found  by  subtraction. 

The  part  of   a   dividend  which  remains  un- 
divided in  an  example  in  division. 
Retailer One  who  sells  merchandise  in  small  quantities. 

to  the  general  pubhc. 
Right  Angle An  angle  formed  by  two  straight  lines  meeting 

in  a  point  in  such  a  way  that  if  both  lines 

were  lengthened  to  cross  each  other,  all  four 

angles  so  formed  would  be  exactly  alike. 
Right  Prism A  sohd  having  four  square  or  oblong  sides  and 

two  parallel  square  or  oblong  ends. 

Right  Triangle A  triangle  having  one  right  angle. 

Roman  Numerals The  numbers  which  are  often  used  on  watches, 

clocks,  etc.,  as  I,  V,  X,  L,  C,  D,  M. 
Root Any  one  of  the  equal  factors  which  produce  a 

power. 

Sales-check See  "Sales-sUp." 

Sales-shp A  paper  showing   the   cost   and   quantity   of 

merchandise  purchased. 
Satisfaction  of 

Mortgage A  document  showing  that  a  mortgage  has  been 

settled  or  satisfied. 
Scale The  relation  of  the  size  of  the  drawing  of  an 

object  to  the  size  of  the  object  itself. 

(VIII-165) 


ARITHMETIC 

Sector Any  part  of  a  circle  less  than  the  whole,  the 

figure  being  bounded  by  an  arc  and  two  radii. 

Seller One  who  sells. 

Semi-circle One  half  of  a  circle,  the  figure  being  bounded  by 

an  arc  and  a  diameter  of  the  circle. 

Septillions'  Period The  period  of  the  9th  rank,   counting  units' 

period  as  the  first. 

Sextillions' Period The  period  of  the  8th  rank,   counting  units' 

period  as  the  first. 

Short  Division The  method  of  division  in  which  no  partial 

dividends  are  written. 

Simple  Denominate 

Number A  denominate  number  which  consists  of  only 

one  denomination. 

Slant  Height The  shortest  distance  between  the  vertex  and 

a  point  on  the  perimeter  of  the  base  of  a 
pyramid  or  cone;  or,  the  shortest  distance 
between  points  on  the  perimeters  of  the 
upper  and  lower  bases  of  a  frustum. 

Speci.J  Working  Unit.  .  .A  unit  of  measure  consisting  of  the  product  of 

two  units  of  measure. 

Specific  Gravity The  ratio  of  the  weight  of  any  volume  of  any 

liquid  or  solid  substance  to  the  weight  of  an 
equal  volume  of  distilled  water;  or,  the  ratio 
of  the  weight  of  any  volume  of  any  gas  to 
the  weight  of  an  equal  volume  of  air. 

Sphere A  round  solid  bounded  by  a  uniformly  curved 

surface,  every  point  of  which  is  equally  dis- 
tant from  a  point  within  called  the  center. 

Square (In  Mensuration.)     The  plane  figure  formed  by 

joining  the  ends  of  four  straight  lines  of  equal 
length,  so  that  they  form  four  right  angles. 

Square (In  Powers  and  Roots.)      The  power  of  the 

second  degree  of  any  number. 

Square  Measure The  table  of  measures  used  for  measuring  the 

area  of  surfaces.  (See  also:  *' Surveyors' 
Square  Measure.") 

Square  Root The  root  of  a  power  of  the  second  degree. 

Stock  Certificate A  certificate  showing  ownership  of  stock  in  a 

corporation.  (See  also:  "Preferred  Stock" 
and  "Common  Stock.") 

(VIII-16G) 


DEFINITIONS 

Straight  Line The  shortest  distance  between  two  points. 

Subtraction Taking  one  number  or  quantity  from  another 

number  or  quantity. 
The  number  from  which  we  subtract  is  called 

the  "minuend." 
The  number  which  wa  subtract  is  called  the 

"subtrahend." 
The    answer    is    called    the    "difference"    or 

"remainder." 
The  sign  of  subtraction  is  — ,  called  "minus." 

Subtrahend The  number  which  we  subtract  in  an  example 

in  subtraction. 
Successive  Trade 

Discounts Several  trade  discounts  to  be  deducted  one 

after  the  other. 

Sum The  answer  found  by  addition. 

Surveyors'  Linear 

iMeasure The  table  of  measures  used  by  surveyors  and 

civil  engineers  in  measuring  distances. 
Surveyors'  Square 

Measure The  table  of  measures  used  by  surveyors  and 

civil  engineers  in  measuring  area. 

Taxes Sums  of  money  which  must  be  paid  by  the 

citizens  to  help  defray  the  expenses  of  the 
government. 

Temperature The  degree  of  warmth  or  coldness  of  an  object. 

Tens  of  MilKons'  Place.  .  The  place  of  the  8th  order. 
Tens  of  Thousands' 

Place The  place  of  the  oth  order. 

Tens'  Place The  place  of  the  2d  order. 

Tenths'  Place The  first  place  to  the  right  of  the  Decimal  Point. 

Ten-Thousandths' 

Place ThjB  4th  place  to  the  right  of  the  Decimal  Point. 

Terms The  several  parts  of  a  fraction,  or  of  a  pro- 
portion. 

Thermometer An  instrument  used  for  measuring  temperature. 

Thousands'  Period Thousands',  Tens  of  Thousands',  and  Hundreds 

of  Thousands'  Places. 

Thousands'  Place The  place  of  the  4th  order. 

Thousandths'  Place.  .  .  .The  third  place  to  the  right  of  the  Decimal 

Point. 

(VIII-167) 


ARITHMETIC 

Time  Measure The  table  of  measures  used  for  measuring  time. 

Times  (X) The  sign  of  multipHcation. 

Total See  "Sum." 

Trade  Diseount An  amount  allowed  by  wholesalers  to  retailers 

so  that  retailers  can  sell  at  Ust  prices  and 
still  make  a  profit.  (See  also:  "Successive 
Trade  Discounts.") 

Transposition Changing  the  order  of  numbers  or  things. 

Trapezoid A  plane  figure  bounded  by  four  straight  sides, 

of  which  only  one  pair  of  sides  are  parallel. 

Triangle A  plane  figure  bounded  by  three  straight  sides 

joined  to  form  three  angles.  (See  also: 
"Equilateral Triangle,"  "Isosceles Triangle," 
and  "Right  Triangle.") 

Triangular  Prism A  soUd  having  three  square  or  oblong  sides,  and 

)  two  parallel  triangular  ends  or  bases. 

Trillions'  Period The  period  of  the  5th  rank,  counting  units' 

period  as  the  first. 

Troy  Measure ; The  table   of   measures   used   by   goldsmiths, 

silversmiths,  and  jewelers  for  weighing  pre- 
cious metals  and  stones. 

Unit ......;;.;;;;....  Oue  person  or  thing. 

Units'  Period; ;  .  ;  ; Units',  Tens',  and  Hundreds'  Places. 

Units'  Place. :  . ;  : , The  place  of  the  1st  order. 

Vertex  (plural , 

"Vertices") The  point  where  two  lines  meet  in  an  angle. 

Vertical. . ; ;  . ,  . , At  right-angles  to  the  surface  of  the  earth. 

Volume; ;  ; .  . :  ; Thc  spacc  occupied  by  a  solid.    (Cubic  Con- 
tents). 

Whole  Number A  number  which  shows  one  or  more  units  or 

whole  things.     An  integer. 

Wholesale?. One  who  sells  merchandise  in  large  quantities, 

and  usually  deals  only  with  other  merchants. 

Working  Units See  "Special  Working  Units." 

Yield* « The  rate  of  income  from  an  investment. 


(VIII-168) 


ABBREVIATIONS  AND  SIGNS 


(Parts  I  to  VIII,  Inclusive.) 


Account Acct.  or  a/c. 

Acre A. 

Altitude Alt. 

Amount amt. 

Angle Z. 

Answer Ans. 

At @ 

Barrel bbl. 

Base B. 

Board  Foot bd.  ft. 

Brackets ();[];   {  }  • 

Bushel bii. 

Cent ct.  or  j^ 

Centi  (.01) c. 

Centime c. 

Chain ch. 

Circle O 

Circ  difference C. 

Commission com. 

Cord cd. 

Credit Cr. 

Creditor Cr. 

Cubic  Foot cu.  ft. 

Cubic  Inch cu.  in. 

Cubic  Meter cu.  m. 

Cubic  Yard cu.  yd. 

Day da. 

Debit Dr. 

Debtor Dr. 

Deca  (10) D. 

Deci  (.1) d. 

Decimal  Point 

Degree ° 

Diameter D. 

(VIII- 


Dime d. 

Discount disc. 

Divided  by h- 

Dollar $ 

Dozen doz. 

Dram 5 

Equals = 

Fifty L. 

Five V. 

Five  Hundred D. 

Foot ft.  or  ' 

Franc f  r. 

Free  on  Board  Cars .  F.  O.  B. 
Gallon  (Liquid 

Measure) gal. 

Gallon  (Apothecaries' 

Measure) cong. 

Gill gi. 

Grain gr. 

Gram g. 

Greatest  Common 

Divisor G.  C.  D. 

Great  Gross gt.gr. 

Gross . .  .  .gr. 

Hecto  (100) H. 

Highest  Common 

Factor H.  C.  F. 

Hogshead hhd. 

Hour hr. 

Hundred C. 

Hundredweight cwt. 

Inch in.  or  * 

Interest Int. 

Kilo  (1,000) K. 

169) 


ARITHMETIC 


Latitude lat. 

Least  Common 

Denominator L.  C.  D. 

Least  Common 

Multiple L.  C.  M. 

Link 1. 

Liter 1. 

Longitude long. 

Merchandise mdse. 

Meter m. 

Mile mi. 

Mill m. 

Mili  (.001) m. 

Minim m. 

Minus — 

Minute  (Time)  .  .  .  .  .  min. 
Minute  (Angle  and 

Arc) ' 

Month mo. 

Multiplied  by X 

Myria  (10,000) M. 

Number No.  or   / 

(#  written  before  a  number) 

One L 

Ounce 

(Avoirdupois) oz. 

Ounce 

(Apothecaries')  .  . .  o 

Peck pk. 

Penny  (plural, 

"pence") d.  ' 

Pennyweight pwt.  . 

Per  Cent % 

Percentage P. 

Pi 7r 

Pint  (Liquid  or  Dry).pt. 
Pint  (Apothecaries')-  O 

Plus 4- 

Pound (Avoirdupois). lb.  or  # 

(#  written  after  a  number) 


Pound  (Sterling) ....  £ 

Power 2   3^  Qf^ 

Quadrant quad. 

Quart qt. 

Quire qr. 

Radius  (plural, 

"Radii") R. 

Rate R. 

Ratio : 

Ream rm. 

Remainder   rem. 

Right  Angle L 

Rod rd. 

Root \/       (called 

"radical  sign")  orrt. 

Scruple 3 

Second  (Time) sec. 

Second  (Angle  and 

Arc) " 

Section sec. 

Shilling s. 

Square sq.  or  □ 

Square  Chain sq.  ch. 

Square  Foot sq.  ft. 

Square  Inch. sq.  in. 

Square  Meter sq.  m. 

Square  Mile sq.  mi. 

Square  Rod sq.  rd. 

Square  Root sq.  rt.    . 

Square  Yard sq.  yd. 

Ten X. 

Therefore .*. 

Thousand M. 

Times X 

Ton T. 

Township T. 

Triangle ^ 

Week wk. 

Yard yd. 

^  ear yr. 


(VIII-170) 


^ 


JB  35864 


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